https://buddypress.org/demo/
If you want to run a site like those examples, your probably gonna need at least a virtual dedicated server. Not sure what exact serverspecs you need though, it’s more based on how many users your expecting.
From the recent SitePoint podcast.
Andy Peatling said:
The BuddyPress.org site is actually over a million users because it’s connected to the main WordPress database/user database. It’s obviously not one million active users, but there is a fair few thousand active users every week, multiple thousands of users every week.
Some of the other standalone websites – testbp.org is actually over about 14,000 users now. TastyKitchen, which is which is another site using BuddyPress for recipes used in cooking, they’re around 20,000, I think.
Check the full podcast on http://www.sitepoint.com/blogs/2009/12/21/podcast-41-buddypress-with-andy-peatling/
https://wordpress.org/showcase/flavor/buddypress/
Maybe developers of those sites can be contacted or interviewed to give more info.
Pretty sure Tasty Kitchen is on a nice beefed up dedicated box. Can’t remember where I read it though.
And it’s a general rule of thumb that only 10% of those registered users will be on your site at a time, on average.