Varies widely. I can speak on experience with the schools that I interviewed at/know about:
Pritzker - scholarship awarded with acceptance
Michigan - scholarships awarded later in the cycle (April)
Penn - scholarships awarded in the spring
WashU - scholarships awarded in the spring with separate app
Vandy - scholarships awarded throughout the cycle, though I'm not sure whether with the acceptance or not
UTSW - scholarships awarded with acceptance
As you can see it varies, but this list is by no means comprehensive.
You can of course ask a school for more money (you'd obviously do this more tactfully than directly asking for a scholarship), but the philosophy of the admissions committee will determine whether or not the school is open to this sort of "negotiation." At some schools decisions are more or less "final:" if you're awarded one, great, if not, tough luck. Others (e.g., Michigan and Pritzker) use scholarships as an incentive to bait people from more prestigious schools. As a result, they'll generally be willing to work with you if money is the only thing standing away of you attending that school (all dependent on the availability of resources, of course).
There are no hard and fast rules, and I know that schools that "don't have scholarships" will make exceptions if you're attractive enough to them. I'm 100% for asking for scholarships or asking for increases - it can't possibly hurt you, but the potential upside is great. Of course you also have to have some leverage. An applicant with a single acceptance can't really bring much to the bargaining table. An applicant with acceptances to multiple institutions that are comparable, on the other hand, can likely succeed in this game.