Is my stash stack pushed to the remote repo? Or is it completely ignored?
I'm just curious if I should tend to it every once in a while to drop some of it to save space on the server.
No. Stashes are local.
$ man git stash:
Use git stash when you want to record the current state of the working directory and the index, but want to go back to a clean working directory. The command saves your local modifications away reverts the working directory to match the HEAD commit.
I wouldn't keep too many of them around locally though. You'll lose track of them over time and they'll become somewhat useless.