Then they could be treated rather like automated crash reports: you could mine them to see if a problem is common, or if someone reports a strange problem you might be able to see whether anyone else had seen it and look for common factors.
Seems hard to make it work in practice without it turning into a place where people would just rant. Maybe if you make the reports readable only by the project's maintainers?
That's a really tricky technical problem to achieve, since many bug reports depend on the user's local environment in order to reproduce the problem, and that could conflict with anonymization.
Making problems public creates incentives for the project to fix issues (since they can't hide their existence) and also provides opportunities and information for contributors to help resolve them (including for purely selfish reasons, to fix issues that affect their own systems too).