That depends, should the state have the right to produce propaganda? No, but it has been legal since Obama repealed the Smith-Mundt Act in 2013. Is that (state-run propaganda) constitutionally protected speech? No. The government is not a person, and by extension, no employee of the government is, either.
Do you and I have the right to produce propaganda? Yes, you are protected by the constitution and should not have a reasonable cause to suspect legal repercussions. You just can't incite a riot, and if you call for violence, you can't name a time or place (or possibly a target? I dunno about that one).