Can individual sharing of a published article form a copyright violation? It is very common for researchers to share published articles upon individual requests, and many copyright transfer statements explicitly allow this.

In the case that the copyright transfer statement does not explicitly allow sharing the published article in any form, is it allowed to share a published article upon request? So suppose that the copyright statement is completely mute on the topic, it simply transfers all copyright aspects to the publisher, except for intellectual property etc. In that case, as the author of the article, is it even forbidden to mail a version to a fellow researcher, or a co-author of the article?

Does this also hold for the pre-print version? For an early draft?
Sending a published article to a colleague, or to someone else who has requested it, or even to a co-author, almost certainly qualifies as distribution, which is one of the six rights reserved to the copyright holder under US law. So yes, technically it would be illegal. There is a fair use "exemption" that could be relevant in some cases, but that varies depending on the specific circumstances.

Whether this also applies to a preprint or to an earlier draft depends on whether those things are included in the copyright transfer agreement.

Of course, as mentioned in the comments, it's unlikely that a journal publisher would care enough to actually make a court case out of a researcher sharing just a few copies of a paper. Honestly, it's unlikely they'll even find out.