Consulting while being a postdoc I would like to hire a postdoc who is working for another institute as part time consultant for my (proposed) federally funded project (in USA). But I don't know if this is even possible for a full time postdoc paid by a private university. Is this a plausible scenario, or should I forget about it?
Having done such an agreement in the past, it's certainly a plausible scenario. A couple things to consider:


Consulting means they can't access their university's resources. If you need that, it'll likely need to be a subcontract, and there are some institutions that allow postdocs to do that on their own, and some that do not.
There are almost certainly rules at the postdoc's university regarding consulting - it can be everything from "don't do it" to "don't let it interfere with your job here" and everything in between. They should check to see what the peculiarities of their institution entail.
While it may or may not be required, the postdoc should almost certainly clear the consulting work with their supervisor.
Non-U.S. citizens working in the U.S. have a different set of requirements, and again, they should check on the specifics of their situation to make sure they're allowed to do so.


TL;DR: It's perfectly plausible, but the specifics vary widely.