In terms of f-stops, how much light is typically lost when shooting through an umbrella?
That's going to vary a great deal depending on whether it's a bounce or shoot-through, what material is used, etc. In general, the range can be as little as 1/2 stop (or less) through to several stops. If you're interested in a particular brand, you can probably find details on light loss in the product literature. Lastolite, for example, will supply the light loss information for their umbrellas.
Saying it a different way, a regular reflected white umbrella (with black cover) is seen to meter 0.7 to 1.0 stops stronger than the same umbrella as shoot-through (without cover), if both are metered at same distance from the fabric. And reflected is slightly softer light too, if at same distance. (Yeah, the shoot-through can be placed closer, and you'd have to use it when that is needed, but that's hard to compare). The black cover doesn't affect the usable light, it only prevents the rear spill.
So that says nearly 2/3 of the light spills out of the back of a shoot-through. The other third is either absorbed or transmitted... How much for which one seems unimportant if we use reflected. :) http://www.scantips.com/lights/umbrellas3.html