Here's my take on it: The second type of privilege should be awarded cross-site, and not the first type. Let me explain why.
Say you have user x, who has started a new account on StackOverflow. They are excited to earn reputation and help people! They earn around 200 reputation each day from honest upvotes and accepts, and without effort, in a month reach around 3k.
At this point, they discover other sites on the StackExchange network and thus create an account in Hinduism StackExchange, Physics StackExchange, Blender StackExchange, Photography StackExchange, and so on.
They first go onto their Hinduism account and click on a question. I'm sure after 1k rep, many people just get that habit of fetching the vote count. But for some reason, they cannot collect it here! After a second, they realize that they don't have enough rep to do so on Hinduism, since each site has a separate rep count.
But then, they think,
Why is it that I can't fetch vote counts? After all, there's no harm in it, and I can already fetch it on StackOverflow.
Now, I know the established user example is not the best, but hopefully, you understand my point that reward privileges to the user should probably apply to all sites.
The second type of privilege (such as close-voting, delete-voting, etc) should not be applied on another site. To earn this type of privilege, you must earn the rep on that site. The reason for this would be because each site has a different topic, and site x would have different rules from site y.
Lastly, these are the reward privileges:
Reduce ads: Since you are already mature on one site, there is no reason to have ads on another.
View close votes on your own questions: Specifically not including the other part, which is to cast close votes on your own questions.
Seeing deleted answers: Only that part! The other part (such as having access to the NATO queue, etc) must be earned by gaining rep on the site itself.
Perhaps internal access to site analytics? I need more input on this one.
If this was possible programmatically, then I feel this would be a very good feature.