Virtually all, if not all, infringement questions can't be answered on Avvo. Your fact pattern is just too abstract and contextless for anyone to guess what these two marks are, and which word you have in common, the order, spelling, punctuation, sound and look of the words in the marks all matter, and you haven't disclosed this, nor should you on this public site.
The one fact that stands out is that you and your competitor offer the same services, and woudl have TMs in the same class, and that seems to make it likely that there might be consumer confusion, which is the crux of a trademark infringemnt question.
You really need to consult a TM lawyer and disclose all the facts about what marks you and your competitor are using and how and since when, and get some specific advice.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
To add to the previous post, while your fact pattern is vague, you should understand that domain names can be viewed as an infringement of a registered trademark (irrespective of the fact the parties reside in different states). The consequences of this can be serious from an economic and practical perspective. Namely, you can be forced to lose your domain name through a Lanham Act complaint and/or a UDRP proceeding, if that is the case. While you have an overlapping word, without further information as to the actual overlap and the nature of the word itself in context of the businesses' brand collateral, it is impossible to say one way or the other that you may have a problem. Hence, if you rely on your domain name as a major part of your company brand presence, you should consult with an attorney familiar with trademark for an analysis of your situation.
I hope this helps.
Disclaimer: This answer is for informational purposes only and does not constitute general or specific legal advice, nor create an attorney client relationship.
In addition to Ms. Koslyn's correct points, a FEDERAL trademark registration (as opposed to a state trademark registration) automatically gives the federal trademark owner rights across all 50 states and U.S. territories to prevent others from using the same or substantially similar marks for the same or similar goods/services (owners of "famous" marks can even prevent uses of similar marks on different goods/services and sometimes even in non-trademark contexts).
In fact, it is incumbent upon a federal trademark owner to prevent use of a confusingly similar mark (especially for the same products and services) - not so much for the company's sake, but to protect the public from confusion.
Although marks have to be considered in their entireties, a trademark attorney would need to know what the shared word is. If it is dominant and distinguishing (e.g., HAIR-DO HOUSE versus HAIR-DO HUT), the shared word may carry greater weight in an infringement analysis.
Since your concern was sufficient enough to ask this question, and because trademark infringement litigation can be very expensive, you probably ought to consult with a trademark attorney to avoid spending more time and money using a trademark that you ultimately might have to forfeit. Re-branding can be an expensive proposition in and of itself (business cards, product labels, website redesign, etc.) Furthermore, if you are in fact infringing the other company's federal trademark registration, you might be advised by your trademark attorney to stop use immediately to avoid incurring damages above and beyond what you may have already caused - just in case the other company sues you.
Your follow-up question: If a company offers organic produce, it cannot use the word 'organic' in its name, such as "Betty's Organic Produce" -- if someone else has already registered the trademark name "New York Organic Farm." Is that a correct reading of trademark law?
Response: No. The analysis is much more nuanced than that. In your example there is nothing in trademark law that prevents the lawful use of "Betty's Organic Produce" in light of the existing "New York Organic Farm" trade name (which may also be a trademark, but maybe not). For lots of reasons.
As all my colleagues have noted, you need to speak with a trademark attorney.
BTW -- I respectfully disagree with Attorney Klosowski's assertion that the federal registration of a trademark precludes others from lawfully using the mark to sell the same goods. Trademark rights are created under the common law of the state(s) where the mark is used. The federal registration of those rights only memorialize those rights and creates certain presumptions and enforcement benefits. The federal registration does NOT confer on the registrant the right to prevent all others from using the mark in geographic areas where the registrant is not using the mark (and, therefore, where the registrant owns no trademark rights).
For example: Company A owns retail mattress stores in Florida and Georgia and uses its federally registered "Keep-In U Deep Sleepin'" as a tag line. Company B owns retail mattress stores in California and may lawfully use that same tag line in California w/o infringing on Company A's trademark rights. Only when Company A starts doing business in California must Company B stop using the tag line -- and if it was the first to use the tag line (but did not federally register it because it either forgot or did not know that it could) then, as the senior user of the mark, it does not have to stop even when Company A starts doing business in California.