You can cut/copy/paste across to/from your screen sharing session. It just does not share the copy buffer automatically. Instead you need to transfer the copy buffer into either direction explicitly by selecting Edit -> Send copy buffer or Receive copy buffer (the actual english text might be different, you will figure it out). Doing so will not copy or paste anything in itself. You then need to paste on the target side.
For example to transfer from your host to your remote machine:
* copy on host machine
* Screen-Sharing: Edit->Send copy buffer
* paste on remote machine
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