Add and option to playlists so that the default playback behaviour is random, this would be on a playlist by playlist option not a system wide setting. This could be done with a check box when editing that playlist. This way when enabled anyone that plays the playlist will have it play back in a random order. This would include people that you have shared the playlist with.
Until this is implemented, if ever, there is a way to create a playlist that automatically randomizes itself every time it is loaded, by any user, which works very well:
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Early 2021 clean-up: implemented (playlists have play + shuffle option)