Please answer for piano specifically.
So classic 3 note jazz voicing to my understanding is Root in the left hand and the 3 and 7 in the right. But what if you have an inversion with 3 or 7 in the bass? Do you still play 3 and 7 in the right hand or does that change?
Laurence Payne's answer is good as far as it goes, but I understand your question to mean that you are playing a solo.
There is no way to give a solid or definitive answer without more real context. But, speaking generically, if you're doing a piano solo with, say, C7/E, your instinct to give the root and seventh to the right hand is probably a good one. It gets more complex if they ask for a note that would normally be skipped, such as C7/G. In that case, since the left hand is given, and the chord must contain its primary color tone and its major mode, you would typically skip the root of the chord.
In reality, of course, it will also depend on where you are coming from and where you are going to. You may be forced to make less typical choices in unusual or tricky situations.
An even more 'classic' piano voicing might be LH 'shell' (3rd and 7th), melodic stuff in RH and leave the bass to the bass player. But that's not the whole answer either.
Remember, the inversion is determined by the bass note - whoever's playing it! If there's just a piano, your main responsibility is to provide the bass line. YOU decide what inversion the harmony takes. If there is a bass, your job is to keep out of his way and do whatever is useful, rhythmically, harmonically and melodically, with either hand, you won't be playing in the range that determines the inversion. If there's a guitar as well you've got someone else's toes not to tread on (and he on yours, though it can be hard to stop a guitarist strumming along :-).
Replying to your comment - for SOLO piano, your RH will generally want to play the tune, whether a tasteful rendition of the original melody or your own improvisation! There are various techniques of adding the harmony 'in the middle', a stride LH, block chord voicings under the melody in the RH, sustained 'thumb notes'... We're (presumably) not improvising Bach-style choral harmony, so we don't have to worry too much about doubled major 3rds, parallel 5ths etc.' although, as in all music, they WILL stand out aurally.
