The question is - how is having a custom website any different ? From an end user (me) point of view, I trust Avaaz as a platform to reasonably not mess with my data.
Personally I hope getting signatures will send a signal to other entities (perhaps more powerful) that helping out with this cause might be beneficial for them. Just because so many people, strangers and friends a like care. Signatures might be useful to fuel other channels like calbear81 is suggesting.
The worst thing in my opinion would be just forgetting people like Bassel and many others in countries with oppressive regimes.
Let's assume for a moment that one campaign to free someone in such situation would be successful. One person. From hundreds/thousands in one country. Probably tens of thousands throughout all oppresive countries. What would be the result? I'll tell you: the interwebs would proclaim it's own great success, that's a small leap but a giant step, blah blah. And happily in their chairs keep clicking, feeling like a hero while per every petition there is a thousand people keep being locked up and tortured and/or killed... Instead of exploring new ideas to act and engaging in more 'on the ground' action to help opressed people (probably the least but something practical you can do from home is keeping Tor running[1]), the web keeps signing up petitions. Taking part in such slacker campaigns is IMO a lose-lose situation, sorry.
[1] https://www.youtube.com/watch?v=GwMr8Xl7JMQ&feature=relm...
btw Tor worked with Avaaz on collecting donations for communication devices, yet: 1) it was a Tor initiative. 2) https://secure.avaaz.org/en/egypt_blackout/ "$25,197 raised so far. Help us get to $100,000". From the frontpage "14,921,795 members worldwide" with a minimum ammount of $15 it gives only about 1700 members who wanted to pay anything (and I can only guess that this number is much lower and there were bigger donations from single entities).
