There is only one thing that crosses sites, that I know of, and that's only partial, so to speak. If you have earned 200 or more reputation points on any one site, then you are given 100 bonus points on any site you join. It will also apply to other sites you have already joined, in that your rep on the other sites will jump 100 points.
Of note, however, is that those 100 points will not enable you to post answers on "protected" questions. Such questions have a note on them reminding users that they require 10 rep points from that site, and that the association bonus does not apply.
Something else that "looks" like it carries from site to site is the ability to chat in the chat rooms for a site. That is misleading, however, since the chat system is connected to the network as a whole rather than each site. So, once you have enough rep to use chat, it's for chat everywhere. [As noted in another answer by Catija ♦ this is 20 rep points, which can be from several sites, not just one, and Stack Overflow, and Meta Stack Exchange are on different servers, so require their own 20 points. Once you get the association bonus above, you'll have the 20+ everywhere anyway.]
As to the badges and privileges and point structure, it's mostly the same. Actually there are 4 different point-to-privilege scales. Most "graduated" sites use one scale, public beta sites (such as this one) have a 2nd scale, private beta sites have a 3rd scale, and Stack Overflow has its own 4th scale.
Private beta sites have the "easiest" scale which is mostly doubled for public beta sites. That, in turn is mostly doubled for the rest of the sites (most graduated sites).
Because of its size, Stack Overflow has an even higher scale, although most things on the lower end (where I get to operate) seem to be the same as other graduated sites. See the answer by Catija ♦ for this correction.
You can see what the point scale is for any site by adding
help/privileges to the URL. So for this site,
https://interpersonal.stackexchange.com/ the scale is at
As far as I know, the badges are the same for all sites - in spirit anyway. There are badges for earning enough votes in a given tag, obviously each site has its own set of tags, but the idea of a badge for answers in a tag is on each site. Though they have the same badges, earning a badge on one site will not give you that badge on another site. Badges incentivize participation on the site rather than the SE network as a whole. (If you update your profile you get a badge for that, Autobiographer, and when signing up for a different site your profile is copied, so you "automatically" get Autobiographer on that site as well.) Most badges are not based on reputation points directly, though some have minimum vote scores on questions or answers. To earn some badges requires privileges, which are tied to reputation points as well, however.