This may vary depending on the field you work in and what kind of papers you're writing, in addition to your personal style as a mathematician. I mean whether you bother to check the details of a citation probably has a lot to do with how much effort it costs you. In my line of work the citations are usually to other papers within my area of expertise and I do in fact try to look pretty closely at what was done and whether it really gives what I need. But there have been times when I needed something outside my comfort zone and was content to take the word of an expert that so-and-so's theorem did the trick.

On the point about propagation of errors, it seems to me that there is a sort of correction mechanism. If a result is being cited incorrectly, eventually it may lead to a contradiction which would then be unraveled by tracing the problem back. Does anyone have any examples of something like this actually happening?

Oh, I should add that some portion of my citations are "courtesy" citations along the lines of "so-and-so did something related". I'm afraid to say in cases like these my reading of the cited paper will sometimes have been very superficial.