Is it okay to get partial funding for a conference from two institutions? Is it possible (and legal) to get funding from both the host university and the university the student is from, for overseas conferences?

To elaborate, let's say I (a student) am going for an overseas conference. My home university is able to fund me (but only partially). The host university is also providing funding (but limited funds, so I am assuming it is also partial funding). Is it possible to apply for both these fundings?
I would say you could apply, yes, but you should mention that you also applied for the other funding. To make sure that you don't run into problems, I would first discuss it with the responsible people at both institutes, as the rules might differ from conference to conference, institute to institute, depending on country, etc.
Then, I would write down a short statement of cost. For example:


Flight:  500
Hotel stay: 350
Conference fee: 100
Meals (if not included in hotel cost): 50


Total: 1000.

For this cost, student ... received partial funding from institute A: XXX and from institute B: YYY.

Make sure that $XXX + YYY \leq 1000$ and then have both institutes sign it. This should put you on the safe side, as you can use it to prove that both institutes knew about the other partial funding and that you weren't "overfunded". Of course it would be best to have an estimate of the costs beforehand and to discuss these things also before you receive money, so once again: Go talk to the people responsible. 

And I hope that it's clear that you should, under no circumstances, get overfunded here. Don't risk it for a few bucks, this might strongly backfire once someone finds out.