Is there any way to find out how many copies of a set were produced? I would like to know how rare some of my older sets actually are.
There is an indication of the rarity of a set in the LEGO Collector's Guide.
So you can see both reviews consider this rating system to be pretty much useless (and the reviews of first editions weren't better) and it doesn't indeed compare to actual figures. However, the fact that they weren't able to secure any is probably a fairly good indication that these figures are not available to the general public. (And knowing LEGO, there's a chance that they haven't kept figures themselves from the distant past anyway.)
So while this is far from an actual answer to your question, the rarity rating of the Collector's Guide is probably the closest you'll ever get. Maybe a future edition will have more, but I wouldn't count on it.