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At the moment it's not possible to message another Spotify user within the app, however, we recommend having a look at this Live Idea suggesting the implementation of a feature allowing Spotify users to message each other.
You can support the idea by adding your +VOTE and clicking on Subscribe in the three-dots menu at the top right corner next to the title. That'll make sure you stay up-to-date with any news about this. It’s also worth knowing that the higher the number of votes an idea gets, the more likely it is for the idea to be implemented.
We always take new ideas and feedback into consideration to improve the app. Here you can read more on how ideas work.
Hope this helps. Let us know if you have any questions.