The above responses are not correct. A reflector, by itself, won't change your basic exposure, it will change your contrast ratios by lightening your shadows. Now, if you bounce an additional light off of said reflector then it most likely will affect your exposure.
If you've got a spot meter that is capable of measuring flash then you can verify my statement. Set up a single light as your main, say at least 45 degrees to your model so that you can get some good shadows to measure. First, take an overall reading using the incident portion of the meter. Next, use the spot portion of the meter to measure and record your subject's highlights, mid-tones, and shadows.
Now, add a reflector to your lighting solution. Use a mirror if you like and place it as close to your subject as you like. Again, take an overall reading followed by a series of readings on the highlights, midtones, and shadows. You will see an exposure change in your shadows, but you won't see anything that your meter will measure in your highlights and midtones. You'll just have more mid-tones. Your overall exposure will remain the same, but you will have less contrast in your image due to the lightened shadows.
It is impossible to achieve a 1:1 lighting ratio with a single light source and reflector. The inverse-square law bears that out. So you can't "add" light to your solution in this fashion. If 500 w/s came out of your strobe, you can't use a reflector to "add" an additional 100 w/s. But you can redirect some of that energy into areas that weren't illuminated by it in the first place.
As to your original question of when do you take a meter reading and why, I take readings before and after. I take my first reading after I've got my basic lighting solution (main and fill) set. I then use a spot meter to check and make sure all important parts of the image are within a 5-stop range and that I'm not blowing out any highlights (2-2.5 stops more than the overall exposure). I pay particular attention to the contrast ratios on skin, because anything greater than a 2.5:1 lighting ratio looks pretty severe. If something important falls outside of that range in the shadows I'll add reflectors or technical (a.k.a. tertiary) lights to bring those areas up.