Keep logs local, and send only errors and noteworthy events to a central server like sentry.io, rollbar, or something else.
If you log everything to a single server, as you noticed yourself, it will become very expensive and difficult to filter out the stuff that you really need when looking for errors or when you try to hunt down a specific problem.
I used sentry.io for a project and it worked well for what it is - error tracking.
But when users reach out with issues, it is useful to look at logs and having them local means I would have to look at each log file per instance which could be on multiple servers.
If you log everything to a single server, as you noticed yourself, it will become very expensive and difficult to filter out the stuff that you really need when looking for errors or when you try to hunt down a specific problem.