How do you measure a social network's success? (number of users signing up)

Lots of questions in one! You might want to split them up for more specific answers.

For the one in the title, the measure I would use is "active" users, where "active" means not just logins but something like "activity on more than one day, and in the past 30 days."

Or, depending on the goal of the social network, more specific things like messages-sent or profiles-viewed. On this site, for example, # active users is nice but # questions and # answers is better because it shows contribution.

Don't worry about the other "social giants." Besides the derth of real data, that data is years old and therefore irrelevant today. Do you think Facebook's data is relevant in a world where most social networks rely on Facebook itself for growth?

The final measure is, of course, revenue. Of course you're probably not worried about revenue yet, which might in itself be a mistake. You should at least have a notion of how you would have revenue, and then you should measure how you're progressing towards that goal. So if the goal is "pageviews for ads," then "pageviews" is your number to measure.

Advertising will only be good once you have a considerable volume. No one will pay for ads in a community that has a couple of thousand members.

You need to reach the thousands (if you are very niche that might be enough). In most cases, hundreds of thousands (and even millions) is the scale that will get you advertisers to pay good money to participate.

As an example, LinkedIn has only started selling Ad space recently, and they've been around for a long time, and grew to millions of users...