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I haven't been using Go modules, to the point where I only have a vague notion of what they do.

I haven't felt the need for them.

At what point do they become necessary? Which pain-points should I be on the lookout for?

Probably at the point new versions of your dependencies break your code. If you have no external dependencies, you should be fine.

It also gives you the freedom to have your projects wherever you want.