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The information they ask you is different per country. (You can check what's your country's required information here.) The information is handled by both the European Commission and by the organizers of this petition, and is given to the member states so they verify every signature (more info here).

I see some countries asking for a lot of info, so if don't want to trust the petition's organizers with personal info or think that the body that has been chosen to confirm your countries' signatures (you can check it here) could do something nefarious with the knowledge of who signed this petition, it's better to not sign it. Otherwise, I'd say go for it.