Stable Diffusion is probably far from the state of the art, but good $DEITY did it open the floodgates! Just watching the open source scene evolving as a bystander is interesting.
What kind of GPU would you need to run this locally? (the training I mean)
Using offload to CPU (with dreamspeed) one can get away with a 8gb gpu. I haven't tried training myself yet, but there are reports of it working with 8gb vram (here for example https://www.reddit.com/r/StableDiffusion/comments/xwdj79/dre...
I have an rtx 2070 with 8gb and it has been working quite well for me. However there are always models that will not fit. For those running on the cpu with potential nvme offload is not that bad. For example a single inference on bloom 7b (30gb of ram required just for weights) on a 32gb ram machine takes about 30s (it has to offload few GB to nvme). This is on zen 3 ryzen and with no gpu use. I can't wait to try cpus that support avx512.
This area is moving really fast, one-click solutions are already being created.
People are also averaging weights of multiple models to create new models based on multiple other dreambooth models, and it works surprisingly well.
The stable diffusion reddit is a good place to see how all of this is developing.