You can customize Mantis to use the statuses that match your process, please see the following article it the manual:

http://www.mantisbt.org/manual/manual.customizing.mantis.customizing.status.values.php

Let me know if this works for you.

It is a great article.

Mainly, it resolves status needs. But...

• Could I allow only testers to change issues into "verified" status ? (developers can't do it, but there are higher statuses for them)

• Could I allow only an external project developer to access only one private issue ? (not all public issues for everybody and not all private issues for him)

• Finally, we have two assignments for one issue: reporter and developer. But the reporter could not be the tester. And we would like to have filters to know who has tested the issue (the tester)

Could I allow only testers to change issues into "verified" status ? (developers can't do it, but there are higher statuses for them)

yes

Could I allow only an external project developer to access only one private issue ? (not all public issues for everybody and not all private issues for him)

This would be a challenge, not sure it can be done

Finally, we have two assignments for one issue: reporter and developer. But the reporter could not be the tester. And we would like to have filters to know who has tested the issue (the tester)

Not feasible, without using a e.g. custom field or a tag to store the tester's name