It is quite common for contractual agreements to include clauses that stipulate that an employee or contractor cannot work for or contract with a competitor (or other entity in the designated field) for X period of time following the breaking of a contract or the conclusion of the contract period.

Fox would have def. had this in place: it prevents the contractee (i.e. Tucker) from working on [FOX] to highlight his fame, reach, etc, and then getting poached by or skipping over to a [competitor].

If Tucker is being paid by Twitter, it means he's being contracted by them, and that could very easily violate such a standard clause.

I'm thinking that he's NOT getting paid by Twitter, currently. But that's just my speculation.