There are three issues here.

1st: “trademarks… in one state.”

“trademark” is generally used to refer to the federal trademark protection for an entity to use throughout the United States. The simplest answer is that the mark holder can then exclude all other users of the mark in the United States.

• first caveat, presuming mark is enforceable: the goods and services have to be similar enough that it would be likely to confuse a reasonable person as to the source of the goods. 
• second caveat, presuming mark is enforceable: the usage has to be similar or not otherwise likely to confuse the audience to whom the mark is used to market. This exception typically does not (IMO) include the scenario of buyers of the goods and services merely being in a distinct state from where the markholder does business. It may include something like the same name not being confusing within the same field, where the mark is generally used to sell generic goods to the public but another entity might use it to sell very specific goods to a very sophisticated buyer. This is easier to argue where marks are not literally identical. 
• Third caveat: questioning the enforceability of the mark. If a mark is issued, the mark can be used to bring suit or challenge the use of the mark or a confusable mark by another entity, but the validity of the mark can then be challenged. This does not particularly help the other entity from being able to use or protect that mark from use by yet-another entity, but it could be useful to prevent the mark from being enforced against a similarly-operating business. Consider “capital city physicians.” Physicians is a term generic to the services provided, and capital city is merely a geographic term. Both are generic descriptions that make it unreasonable to allocate to a single entity.

2nd issue: state marks

Several states have specific “state trademarks.” These are generally intra-state limitations on imitation, and are not enforceable against users in another state, even if the business does business in that foreign state, without a foreign mark. However, this does nothing to overcome the enforceability of the mark under federal trademark, which takes precedence over state protections, under the supremacy clause of the constitution, if not also by literal recitation in any particular state’s law or precedent, regarding the conflict.

3rd issue: ”what happens, if …uses…”

There is no automatic mechanism to enforce protection of otherwise enforceable marks. IP-Rights-holders still have to actually take steps to enforce the mark, to effect exclusion of use. This means notifying the infringer of their use in conflict with the entity’s right to enforce, and filing suit, if the use does not stop.