I believe the rules for signing petitions strike a good balance in not requiring signers to make their name public even if real information must be submitted. This should be extended (as an option) to people opening petitions as well.
You may be interested in these active petitions
Not everyone is comfortable going public on a potentially controversial topic, and there's really no need to require it. The downside is a likely strong chilling effect on what petitions are opened -- likely much stronger than the requirement of submitting real details.
I was in fact planning to open another petition, one where I wouldn't mind the government etc. knowing who I am. The broader public is another matter though, so I've changed my mind.