He can certainly send private emails, Slack messages, or even Twitter DMs. What he can't do is publicly broadcast his tweets to the world, and then exclude people from them on political grounds.
Pretty sure he can't and if he is, he better be complying with federal records laws or else he'd be living a double standard. You may recall him blasting another, very popular candidate in the 2016 election for using personal email to conduct official business. Using slack / private email / twitter DM's is basically the same damn thing.
What if he blocks them on non-political grounds. Maybe they made fun of his mother, or dog, or threatened to throw dirty socks on his lawn, should be be able to block them then?
Are they really public in that case?
The President can share information with 1 person, 2 people, 3 people, or 336m people (the number of Twitter users). But he can't share information with 335m people?
That said, excluding certain people from official communications based on their opinions might be illegal too: https://www.nytimes.com/2017/02/28/us/politics/white-house-b...