It actually varies by species! Penguins, on one extreme, have zero hollow bones, keeping a heavy, marrow-filled skeleton to help with deep diving.

For most other birds, the skeleton is actually quite marrow-rich when they are just hatchlings. As they grow into adulthood, the skeleton gradually becomes pneumatized, with air sacs displacing red marrow in many bones. However, this displacement is not 100%...small spots of marrow persist into adulthood in many bones that researchers would otherwise call "pneumatized" or "hollow".

A few bones tend to escape pneumatization entirely, especially the femur and the tibiotarsus. In most birds these red-marrow-filled leg bones will make up the bulk of blood cell production. Other common red-marrow sites in birds are the radius/ulna in the wing, and certain sections of vertebrae (especially the caudal vertebrae in the tail). Many other bones will retain just a small amount of marrow in some birds, and none in others, so there is variation from species to species...but in general those three sites (fibia/tibiotarsus in legs, radius/ulna in wings, vertebrae in tail) will play a big role in making red blood cells for most birds.