How to replicate growing files?

I need to replicate a bunch of continuously growing large log files from a network file share.
Can't use rsync, because I don't have ssh access to the server.

Is there any existing Linux tool that can do incremental replication based on comparison of source/target file sizes only? Assuming that the already copied portion of a file is not going to change is okay.

Caveat lector: large, monolithic, non rotated log files are generally a bad idea. If you do keep such logs, I'd advise you to think long and hard as to, why you consider that the best solution (spoiler: it likely isn't).

Depending on how you approach your current problem of duplicating logs to two different targets, the best way would be to simply configure the log-sending endpoint to have two different targets, this would speed things up and simplify managing log files on both targets.

Given the current state of things in your question, il assuming that's a no-go either. So as a last resort, dd has options to start reading from a particular offset, and any custom made stat, open, seek sequence would yield what you aim for with little to no effort (eg. Python would do that in <20 lines).

You can backup network drive to Google Drive, which will only upload the files that have been updated everytime.