Unless the business informs you that you have been included in a composite state tax return filed by the entity, the individual receiving the K-1 is responsible for the respective state tax filings.
Having said that, you need to review each state K-1 and then the instructions for each of those states. Every state will have different filing thresholds. Additionally, some states require that the entity remit withholding for nonresident taxpayers. If that is the case, you should be informed of this on the respective K-1 or notes to the K-1. These amounts are only estimates and are typically based on the highest state tax rate. The only way to know if you are due a refund for that state is to complete the return and then file if a refund is due.
*A reminder that posts in a forum such as this do not constitute tax advice.
Also keep in mind the date of replies, as tax law changes.
