I think it's a simple power inversion. Conservatives banned liberal speech so liberals advocated free speech. Now liberals ban conservative speech so conservatives advocate free speech. Does it have to be more complicated than that?
Sure it's hypocrisy, but it's also simple self preservation to prefer more freedom when out of power and more control when in power. I don't know of evidence that this effect is stronger for one side or the other.
I generally agree that tolerance of "free speech" is a highly-dependent thing, on either side, with hyporcites too numerous to name. And that, much like "states rights!", it's a battle-cry embraced when one side is losing some larger cultural or legislative fight.
However, there are more state-level laws aimed at curbing speech coming from Republican-controlled legislatures than Democratic ones, and that does seem to be a difference in kind.
> state-level laws aimed at curbing speech coming from Republican-controlled legislatures than Democratic ones
Examples?
https://capitol.texas.gov/tlodocs/871/billtext/html/SB00005I...
http://www.leg.state.fl.us/statutes/index.cfm?App_mode=Displ...