Yes, this can be done. However, boards with components on both sides will cost more to build up because it requires extra steps. As you say, you don't want the chips on the bottom to fall off while the ones on top are being soldered.
This is usually dealt with one of two ways. When the part has low weight to solder pad area, then it will stay in place just from the surface tension of the molten solder. For heavier parts, usually a kind of glue is used.
Double-sided boards are common enough, but when doing something like this it is still a good idea to talk to your assembly house first to get some idea what their processes are and what you can do on your end to make things simpler for them, and therefore cheaper for you.