This Nightbot custom command allows your chat to save currently playing song to their own list that can be viewed later.
It'll ask your permission to read your queue and channel's name - nothing else is needed.
If something doesn't work, PM me at twitch (Enbis) or discord (Enbis#3840). Source code is here, you can upload it to your own host if you have one. It's only better for me since my hosting provider is cheap and slow - not sure it'll handle it if more than a few people would use this.