Employment in the state of Colorado is "at will" absent an express agreement to the contrary. At will employment can be terminated for any reason not amounting to discrimination on the basis of a legally protected trait (race, religion, gender, etc.) or retaliation for engaging in certain forms of legally protected conduct (filing a wage claim, taking FMLA leave, etc.). It doesn't matter whether the basis for termination is fair, reasonable or even true. This means that an employer is your circumstance could certainly terminate an employee because you believe (rightly or wrongly) that the employee is in violation of a non-compete with a previous employer.
Termination would typically be the safest option, too. Though it is unlikely, once you become aware of an employee's non-compete with a previous employer and you continue to employ that individual, you are potentially exposing yourself to a claim for tortious interference with a business contract. It is rare that employers are sued on this grounds but it is certainly possible.
As for the employee responding to the cease and desist, no law requires someone to respond to a cease and desist letter. It's just a letter. It may "state" deadlines but those deadlines are entirely self-imposed. The employee doesn't need an attorney to respond to the letter, nor do they need to respond within any particular period of time. Of course, the risk is that if they don't respond or take the action being requested that they will be sued. Then, once they are sued, they will need to retain a lawyer and there will be strict deadlines moving forward in the litigation.
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