In my mission I have one player with backpack and supplies in it. But when some other player wants to access content of his backpack, he sees only empty space.

That is because I'm adding backpack to that player only on his PC, and the content of that BP is again added only on the BP's owner PC. So other players basically don't have created content for that BP -> only empty BP is seen.
I know that BP's content could be streamed trough scripts, but I think it might be engine's function to stream contents of backpack across all clients on the server.

(Or is there something wrong script-side, which is causing BPs not to share their contents with all clients?)