I wish I still had a log of the conversation or remembered the exact problem that led up to it, but it involved a simple amend totally screwing up my repo, and I've avoided it since.
Amending your commit in your own feature branch or fork isn't a problem. I often amend commits to make a cleaner git history to ease PR/code review.
git checkout master
git checkout -b 10-new-feature
# Make changes
git add example.code
git commit
git push origin 10-new-feature
# Oops... left a debug message, fix it.
git add example.code
git commit --amend
git push -f # Rewrites the remote history with your local modified version
Amend after push, I could see. You commit, push, amend, push, but after your first push someone else pushes... then you have two heads. I could see that, but I'm guessing you'd mention that if it were the case.
If one were to attempt avoiding amend you could always make all changes in a feature branch and rebase this branch onto your target afterwards, squashing commits.
Once other people can be expected to have done work based on your amended or rebased work, you can expect some annoyances (or worse). If you haven't pushed yet (or if you're really, really sure nobody's basing further work on something you've recently pushed), I say go crazy and amend and rebase to your heart's content.