It depends upon what exactly you are asking in reference to what type of opposition would work and what would be the geographic scope of that opposition.
In regard to federal trademark oppositions, the registration of a federal trademark can generally be accomplished through two manners. First, by filing an Opposition Proceeding with the U.S. Patent and Trademark Office's internal court known as the Trademark Trial and Appeal Board. Second, by challenging the ability of the other mark to register through a larger federal court action filed with a U.S. District Court that retains jurisdiction over the owner of the applied-for trademark.
Should you be successful through the first avenue (Trademark Trial and Appeal Board) that may preclude the opposed applicant from again attempting to register the same trademark with the U.S. Patent and Trademark Office under a doctrine known as res judicata. Note, however, a win before the Trademark Trial and Appeal Board only involves the registration of the trademark. It cannot stop the actual use of the trademark in commerce.
For that the second option would have to be considered - filing an action before a U.S. District Court that retains jurisdiction over the owner of the applied-for trademark. Under this course of action you would have the broadest available remedies including, but not limited to, stopping the other's trademark from registering before the U.S. Patent and Trademark Office, stopping the other's use of the mark, and monetary damages. Note, should you win the doctrine of res judicata could also apply here thus preventing, or limiting, the issues from being re-litigated in the future.
Regarding state proceedings, typically state proceedings, either before a Secretary of State or Corporation Commission, can only affect those rights in that one particular state. Likewise, state court proceedings can only affect rights and the ability to use a state trademark in a particular state. So it is very limited in the scope of relief it can provide.
Your last comment indicates you are possibly interested at looking into the possibility of splitting up geographic areas between the two parties. If the other party retains a federal registration of application on file with the U.S. Patent and Trademark Office this could be accomplished by and through a concurrent use proceeding which seeks to divide federal trademark rights in specific geographic areas to diverse trademark holders. Of note, this is possible but the rules for geographic division among owners and the procedure therefore are very specific and I would suggest you contact someone with experience in the area to see if you specific set of facts qualifies you to entertain the action.
Hope this helps.