Long story short: at the moment, no. You can, however, use private tags and a private page to effectively see and manage new submissions.
Short story long: We’re in the middle of overhauling how submissions (and reviews and management) are going to work, and notifications of submissions will definitely be a part of that work.
Any specific thoughts on what you’d like to see? The direction we’re trending at the moment is that few things that will happen on PubPub will truly be urgent (loosely, need a response within 24 hours), so we’re trying to reduce the sheer number of alerts we send and give users more control over frequency and so on.