As it is, the last part can only be understood as “les affaires ne se regardent pas entre elles”.

I think the only correct way of saying it would go as follows:

“J’imagine que « se mêler de » désigne le fait de fourrer volontairement et indûment son nez dans les affaires d’autrui qui ne nous regardent pas.”

Not sure I can provide a rigorous explanation, but here are my thoughts.
Often the English indefinite “you” is translated by the French impersonal “on”. And this pronoun acts similarly to “nous”. For instance, you could have used “fourrer notre nez” instead of the more general “fourrer son nez”. But when it comes to the direct-object pronoun, “nous” is the only option there is for “on”.

Hope it’s helpful

PS: it’s just a personal opinion, but you could drop “autrui” to be more idiomatic and simply say “dans des affaires qui ne nous regardent pas”