Any experience with turning site-wide backlinks into index-page-only or nofollow?

Got a few of these kinds of backlinks from other webmasters, and I'm wondering if they're ruining my SEO.
They are from good sites (good SE rankings and traffic), from "Friends" blocks that have about 20 links leading to various domains, one of them mine.
These are site-wide links, meaning that they appear the same way on 1000s of pages on the given domain.

They are also traffic links for me, bringing in a low but steady amount of good visitors.

Lately my SE rankings have been…

Any experience with turning site-wide backlinks into index-page-only or nofollow?