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.
Rarity Rating. This would be more useful if it represented actual production or sales figures instead of market trends.
The rating system is the same, and unfortunately the explanation (do you still remember? "This rating evaluates the rareness of the certain item (and not its value!) considering a complete item in very good condition including building instruction, all stickers AND box (if applicable). Markets like eBay, BrickLink and other sources have been considered for this evaluation - on a global level."[copied from my feedback to the 2008 edition review]) has not been extended, highlighted or moved to a place where it will be less ignored. Additionally, this rating will always give room for discussion, especially as it would require continuous updates.
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.