I am an independant, and I study a lot of politics. Typically Democrats want to restrict rights (although sometimes for very good reasons). Without a doubt Democrats restrict the freedom of speech more, but yet it is not restricted at such an extreme that we should even worry about it. This shouldn't even be an argument. Many of my friends who are democrats agree that their political party supports restrictions on society. (But once again, these restrictions are for a GOOD reason most of the time)
Well I mean you can see it lately when the White House won't even call the Taliban terrorists which is strange. The progressive/liberals are the main ones calling any racist they don't agree with. And since they don't agree on anything it can get insane. They have a Politically Correct term for everything. That's why I stay a centrist because wading to that mess is crazy. After the Paris attacks they called for stopping free speech against Islam to prevent terror attacks which make no sense when terrorists hate the west.
The Democrats attempt to place restrictions on people to protect other people. Look no further than the grisly spectacle of political correctness. They attempted to make it mandated to be polite. Gone are the days when you can do the "you are all worthless" speech from Full Metal Jacket. Someone will get offended.
Democrats are the inventors of laws and restrictions, in their minds to help the country be a safer place. However, whether it is your opinion that Socialism is a good thing or bad thing, Democrats are more on the Socialist side of politics. It was a Republican president who spoke AGAINST the government.
I believe that neither party restricts your speak intentionally, but Democratic political leaders restrict freedom of speech less than republicans. One case is that of Net Neutrality, which is something preventing internet providers give specific people and companies better than others. This also gives the internet providers the capability to slow down internet speed for people who are not paying, and it would make it unbearable to go to some websites because of Internet speeds, Making many companies which do not have a lot of money get less business (for more see http://en.wikipedia.org/wiki/Net_neutrality). President Obama is against net neutrality, yet republicans are not. This is just one of many examples of real-world evidence proving my claim to be relatively correct.