If you’re on Linux, I found Gummy to be the closest to Overleaf’s constant recompilation. My default has always been TexStudio, it has a good UI, but you can also use a VSCode extension. These are all just editors, though. You’d also need to download LaTeX locally. On Windows, that’s MikTeX, on Mac it’s MacTeX, and on Linux texlive is usually already installed, but you may need to install packages. On Debian-based distros, they’re grouped into collections like texlive-science.
I will say that I’ve helped friends who were very used to overleaf to a local editor, and they were quite frustrated that TeXStudio wasn’t exactly 1:1 with the overleaf UI. Please know beforehand that if you’re expecting to be able to do things like open images in the TeX editor to check on them before inserting them, that’s not gonna happen.