Both. Use imgur or whatever service you like to actually share so you don't have to worry about bandwidth and hosting. Just also keep a local copy of everything, doesn't need to be a beefy server. Just don't let a 3rd party online service be the only copy.

Wasn't it that Imgur will allow only registered and verified accounts to post in the future?

If the traffic plan allows it: self-hosting images is the way.

I do both, at least in the context of linking say screenshots to others.

If I'm linking a very "casual" or non important screenshot then I'll just toss it up on my server and link that copy.

If it were somewhat important, then I'll upload it to my server and imgur and link the imgur copy, just in case my server goes down - doesn't stop imgur from deleting it either but if I get hit by a train and am made unavailable then imgur is probably the best bet.

If it is critically important then I try to avoid screenshots / pictures all together and just describe it in text because that has the best chance of persisting. And if the text disappears, then at that point the image is probably not as useful either.

One other thing to consider is that if you goal is to keep the image up for 5, 10, 20 years, there's actually way more chances it happens on imgur than on your self hosted webserver : one day you'll be bored of maintaining it, and that'll be that. A good part of the web from the 2000' was deleted because we were hosting our blogs on our own servers, until we didn't. In an ideal world, we would have a p2p web, where content is distributed by users of the site and it stays up for as long as one person seeds it, like the Beaker Browser tried to do with dat, but this is far from what most people use today. For now, the power of self-hosting is more suitably directed at providing apps you will use for yourself than to publish things meant to stay out in the world. Unless of course you're ready to commit to that, knowing from the get go it will be a challenge.