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I am sorry to hear about this situation.
1) Should we respond to the competing company's letter and, if so, with what information? E.g., should we let them know that Employee A did not sign the agreement and thus it is inapplicable to him (and that, if they believe he did so, to provide a copy)? What should we respond with respect to Employee B - should we say we are taking measures to ensure his compliance?
It is up to your company whether to reply or not. At this time, this is merely two private entities talking. It would make sense to reply, however. Failing to reply may give an unintended idea that they are being ignored. If so, then your company (let us call it "ABC LLC") would want to request copies of both of the documents for A and B. If the other company (say "Prior LLC") fails to provide either, then as as ABC LLC is concerned, no such document exists.
If B did sign it, then if they are completely prohibited from working at this time in their position, then ABC LLC has a choice whether to fire B, or not. If B is not fired, Prior LLC can sue B for breach of the contract. It is very seldom that the prior company would also sue the current company, they mostly focus their relief on the former employee, although they can technically try to sue ABC LLC for "tortuous interference with contract."
2) Should we take measures at our company to ensure compliance with the agreement? If so, what measures?
The onus is on the ex employees to follow the contracts, not ABC LLC. ABC LLC did not sign anything. So if A & B do not follow the non-competes/disclosures, Prior LLC can sue them. Again, Prior LLC can also sue ABC LLC for "tortuous interference," but this would be unusual.
3) Is our company really even exposed to any legal liability here, as we did not sign the agreement (but rather our employees did)? Should we even get involved/respond?
See prior answers.
"It has long been held that a stranger to a contract may be liable in tort for intentionally interfering with the performance of the contract. (Lumley v. Gye (1853) 2 El. & Bl. 216 [118 Eng. Rep. 749]; Imperial Ice v. Rossier (1941) 18 Cal.2d 33 [112 P.2d 631], and cases cited.) The elements which a plaintiff must plead to state the cause of action for intentional interference with contractual relations are (1) a valid contract between plaintiff and a third party; (2) defendant's knowledge of this contract; (3) defendant's intentional acts designed to induce a breach or disruption of the contractual relationship; (4) actual breach or disruption of the contractual relationship; and (5) resulting damage." (See Seaman's Direct Buying Service, Inc. v. Standard Oil Co. (1984)
By giving ABC LLC notice, Prior LLC may be asserting its rights to pursue ABC LLC for tortuous interference. But again, in the majority of claims, prior companies pursue the ex-employees, and not the current company.
4) With respect to Employee A's girlfriend, are there any concerns with hiring her? If we do hire her (which we really want to asap), should we disclose this to the competing company? He says he never signed the agreement, and he also says he never persuaded her to join our company.
No. Non-competes do not transfer to family/acquiescence/partners.
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