Posted 10 messages in chat that were starred by 10 different users
Is it possible to check my progress on this badge, and if so how?
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Chat statistics aren't available on data.stackexchange so you can't use it to track it. As far as I know there's no easy/automated way to check on chat-related stats.
Unless they build some badge progress indicator for this (which is unlikely), you'll never be able to check the status. Chat stars are kind of like upvotes: anonymous. There is no way (and probably never will be a way) to check who has starred any of your messages, so there's no way to figure out your progress.