You know that a judicial proceeding has been filed against you when you are "served" with notice. In fact, legal proceedings cannot generally proceed without somebody swearing that you were served notice.
If a criminal complaint is being pursued against you then you might also learn of this fact when an investigating law enforcement agent contacts you to question you or arrest you.
"Filed a case" could mean all sorts of other things. For example, it could be that they filed a police report, or filed a complaint with some company or non-law-enforcement entity. It could be that they have in fact filed a claim in a court of law and whoever is serving process just hasn't been able to find you. In any of these events I don't know of any way that you could proactively determine that without knowing exactly where and how the "case" was "filed."
Actually, if a criminal complaint was filed against you and approved then a court in another state could have issued a warrant for your arrest. Contact your local police and they should be able to do a nationwide search for open warrants on you.
Police will not typically release information on "open investigations." So even if you knew the exact agency where it was filed they may not tell you anything. If they decided not to investigate it then you might have a right to request the complaint under open-records laws – that depends on the state and the agency.