Since a lot of foreigners are talking about this one and I am in a position to read the Danish language, it seems worthwhile to recap it here. The Danish website for the report is here. There is no English language summary that I can see.

The background for the report, Google translated (with manual fixes):

The Freedom of Expression Commission was established in December 2017 and has been tasked with describing the framework and general conditions of freedom of expression in Denmark, the historical development of freedom of expression in this country and the social conditions that are of significant importance for the terms of freedom of expression. The Commission has looked at experiences in other countries, as well as looked at whether international tendencies affect the expression of freedom of expression in Denmark.
The conclusion summary for the press:

The Freedom of Expression Commission concludes that freedom of expression currently has a good framework and conditions in Denmark. Freedom of speech rests on a solid foundation, which the Commission does not see reason to change fundamentally. However, in the Commission’s view, freedom of expression is facing a number of challenges which need to be taken seriously. Among the issues that the Commission particularly emphasizes are security of expression, which includes: covers that individuals through e.g. Harassment, violence or terror try to prevent others from expressing themselves and participating in the public debate.

So, unsurprisingly, it concludes that everything is on fire fine. The part that everybody dissidents are talking about is from the survey the Ministry of Justice conducted on support for freedom of speech. This survey follows the typical approach seen in many studies of freedom of expression in other countries: one asks about general support for freedom of expression (most people are positive), and then concrete examples of controversial expressions (lot of people are suddenly not fine). These are contradictory results, and often used to show that voters are irrational.
The committee that produced this consists of:

I am not familiar with these people except for Mchangama and Fleming Rose, both of which I approve of.

To return to the results of the survey and the included experiment, they are presented in the 84 page summary report here. The translated summary:

This report deals with freedom of expression in Denmark in 2019. The report is part of the of Freedom of Freedom Commission’s work on assessing the framework and general conditions of freedom of expression in Denmark. In connection with this work, the Commission has decided to conduct a questionnaire survey which highlights the attitudes and experiences of the people with freedom of expression. This report contains analyses of the responses to this questionnaire survey, which have been sent to a representative sample of the Danish population aged 16-74 years. In addition, extra responses have been obtained from selected groups in the community. This applies to young people aged 16-25, as well as immigrants and descendants from six selected countries, which in the report are divided into (primarily) Muslim countries and other countries. In addition, separate questionnaires have been sent out to politicians, debaters, journalists and artists. The survey was conducted during the period April to June 2019. The purpose of the report is to map different groups’ attitudes to and experiences with freedom of expression in Denmark.
Selected descriptive results
Overall, there is a high level of support for freedom of expression in Denmark among all groups in the study. However, if utterances have consequences for others, support is limited considerably. The support for freedom of expression is also seen to vary depending on who is speaking and who is being talked about. The former appears to explain much of the variation in the support for freedom of expression. As far as employees’ use of freedom of expression in the event of criticizable circumstances in the workplace, there is generally broad support. This is especially true of public servants. Overall, however, there is an viewpoint that people generally have the opportunity to use their freedom of speech by participating in the public debate, but also that at times they or others on different grounds refrain from expressing themselves in the debate, even if one really wanted to.
The population (representative sample of the population in Denmark aged 16 to 74 years).
The population rates importance of freedom of expression for Danish society highly. However, there is a tendency for the support for freedom of expression to diminish, the greater societal consequences the utterances have. 51% of the population believes that ordinary citizens should be able to express themselves, even if it can hurt or offend other people, 40% believes this if the consequence is that it could threaten the unity of Danish society, and 22% if it could threaten the security of the state. There is a tendency to see greater support for the right of ordinary citizens compared to politicians and the media.
Overall, the population has broad knowledge of restrictions on freedom of expression. The examples given in comments that would typically be illegal to post in an open post on Facebook are at the same time those who think the largest proportion (between 51 and 84 %) think are generally illegal. In general, there is also agreement between what the population thinks, which – and believes should be – is illegal. However, there is a greater proportion who believe that the comments should be illegal than the proportion who believe they already are. As regards the population’s perception of whether certain statements concerning religious issues should be included in the same freedom of expression as other statements, see a larger proportion think it should be illegal to argue against the introduction of Sharia law in Denmark (62%) than the proportion who believe it should be illegal to criticize Islam (18%).
The population of 2019 is generally more tolerant of the right of selected population groups to comment in the public debate compared to a 2006 study. In 2019, there is however large support that two groups, neonazis and Muslim extremists, who argue for the implementation of Sharia law in Denmark, should not be covered by the same freedom of expression as other groups.
There is a relatively large agreement that people generally have the opportunity to participate in the public debate on both the social media and the traditional media. 74% respectively. and 71% of the population believe this. Regarding own participation in the debate, 42% participated in the social media and 23% in traditional media within the past six months. At the same time, around 70% state that they believe there is something that can make people refrain from participating in public debates on the two types of media. The share that, at some point in the past six months, has even refrained from expressing themselves, even though they really wanted to, is somewhat smaller, namely 40% in terms of social media and 23% for the traditional.
Overall, a larger proportion of the population experienced being exposed to deliberately untrue stories on social media (61%) and in the traditional (58%) than the proportion who are most concerned about this (44 and 29%). Both the exposure to untrue stories and the concern for them are greatest on social media.
A greater proportion of public servants have experienced critisizable conditions on the job, which they believe the public should be aware of, compared with private employees (29 and 13%). It is also seen that there is greater support for public servants to comment in the public on critical matters regarding their work-space conditions for private employees.

Young people (16-25 years)
Young people’s attitudes to and experiences of freedom of speech typically do not differ from that of the population as a whole. The few places that see differences are the tendency that young people are less restrictive than the population.

Immigrants and descendants from Muslim countries and other countries
In general, immigrants and descendants of Muslim countries are more restrictive of freedom of expression than the population. They rate the average importance of freedom of speech high, but significantly lower than the population as a whole. In comparison with the total population, less support for freedom of expression can also be seen when this can have different consequences. In general, responses from immigrants and descendants from other countries [i.e., immigrants from non-Muslim countries] most resemble the general population [mostly the Danes]. In some places, however, their responses are more reminiscent of the responses of immigrants and descendants of Muslim countries. This is especially true in relation to the belief that there is something that can make people refrain from participating in the public debate on the social media in the traditional media and in relation to abstaining from it themselves.

Politicians, debaters, artists and journalists
Overall, politicians, debaters, artists and journalists are the least restrictive [most pro free speech] in the investigation. The groups’ responses are very similar to each other. On average, the four professional groups assess the importance of freedom of speech more highly than the general population – even if utterances have consequences. It can be emphasized that a greater proportion of the professional groups themselves participate in the public debate. At the same time, during the past six months, a large proportion of participants have refrained from participating, even if they really wanted to.

Sex
There is no clear picture that there are differences between men and women’s attitudes to and experiences of freedom of expression. In places where differences are seen, women tend to be more restrictive than men.

Education
There is a tendency for attitudes to and experiences of freedom of expression to vary across the educational levels of the population. There is a picture that people with upper secondary education and university degrees – and partly also people with medium-long education [so called professional bachelor degrees like teachers, electricians etc.] – are less restrictive than people with primary education and vocational education. However, these trends are not unique or pervasive for the entire study.

Income
There is no general tendency for attitudes to and experiences of freedom of expression to vary across the income level of the population. The few places seen differences tend to be the second highest and highest income quartile less restrictive than the lowest income quartile.

So to recap, more pro free speech groups:
  • Younger people
  • Danish people compared to foreigners and their children (second generation immigrants). Muslims particularly low.
  • People who work in journalism, artists, public debaters, and politicians
  • More educated people
  • Richer people

Some key figures with translations

Overall agreements and beliefs about current law

By groups of interest

Islam/Muslim matters

This is the most important result that people are quoting. It’s based on the design of showing people, at random, a list of statements, and asking how many should be illegal. This way, subjects do not have to out themselves for their specific beliefs. The estimated support for making something illegal is simply the difference in the counts between the control and experiment groups. This design has been widely used to avoid socially desirable responding on tough questions.

Freedom of speech for who?

Anti-semitism / critical of Jews