By beta reading, do you mean beta in a similar sense to software, so like draft reading/readers?
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Yep! Basically a beta reader will read a written work (in this case fanfiction) and offer feedback on what can be improved, ranging everywhere from plot to characterization. Though some betas may specialize in only one or two components that go into a good work.
This is a guide for beta reading for original fiction, but the same principles apply!
When I was writing fanfic and had a beta, I sent them a word doc in something Google, and we'd discuss it over Skype.
As Skype is dead, and no one seems to like me using open office, I'd probably just post with no beta.
Which I did do for a long time! I feel they were worse though. More rambling.
For conversation with beta readers, I usually do DMs over the platform where they usually do their beta work stuff, which these days tends to be Discord but I've had some cases of PMing over topical forums's message systems.
For editing docs, my usual workflow is I set up a cryptpad or etherpad version of the document for them to edit and annotate and then we compare notes. Other times I use my own hedgedoc instance when what I want is light notes or minor fixes, or I just e-mail the document over.
I've been looking for some sort of fandom-established privacy-respecting Google Docs alternative but so far no one seems to have done it, which is strange considering using Google for that kind of stuff give s enough ammo to some evil companies (ahem... Nintendo) to sue two or more people in one go.