Are there any clear identifiers in speech that people from the former republics make / have / use when theyre speaking? For instance, I have noticed that certain republics pronounce "что" like "чи-то" "шо / шё", and others trill/roll their r's more forcefully and longer than "standard" Russian, per se.
Ukrainian Russian is the only clearly identifiable one, a language variety unto itself somewhat like Irish English (and not to be confused with the Ukrainian language in the same way you also have an Irish language, although Ukrainian is better compared to Scots in terms of similarity). A significant number of people from Kazakhstan, including ethnic Kazakhs fluent in Russian, have Southern Russian vowel patterns (сказал
[skə'zal] rather than
[skɐ'zal]), but that's not consistent. Any other differences would register as a foreign accent or, if they're minor ones, pass under the radar.
It depends how detailed you want to be. Even a native speaker will be surprised at the variety of dialects and nuances, opening a linguistic atlas.
The "чё" or "шо" that you mentioned are quite prevalent in speech in some regions of Russia per se. But the distribution is not even, and also depends somewhat on the education/social level, as these variations are often considered improper by purists, if not outright wrong.
So it's not always possible to make a "clear" distinction that you want. That said, some dialects are easily identifiable, such as Ukranian (Ukranian Russian) by their fricative Ukranian 'Г' and many other characteristic features.