Usually teaching assistants or research assistants will be hired by money from the department or grants, but can they be hired by a direct pay from a professor's salary?
Probably not, though professorial funding can get tricky. In the US, most departments or colleges handle the funding of TAs directly. They are normal employees and are (more and more) entitled to benefits. If a prof just took money literally out of their own pocket and paid a TA, this would circumvent tax withholding, the purchase of insurance benefits, etc. TAs need to be employees of the university. At the same time, since they interact with students and appear in the classroom, TAs need to have official status with the university. Since direct payment by a professor would bypass this, I doubt very seriously that it would be allowed.
However, once a professor has a grant, money becomes more flexible. It's entirely possible for a professor to be paid 9 months by the university and end up funding more than 3 summer months through grants. NSF has its 2 month rule to try to prevent this, but through multiple agencies, a lot is possible. Agencies allow Principle Investigators the flexibility to move salary money between personnel within a grant almost at will, so it could be that a prof could take one of those months of their own salary from the grant, and spread it across several (at half time) for a student. This is pretty common. It wouldn't be direct, the student would be paid by the university and have official GRA status, but it would, sort of, come out of the prof's pocket, depending on your point of view.