In general, there is no "reasonable expectation of privacy" in a private employer's workplace (except in a rest room). Employees of state or federal agencies have marginally greater protections.
So, if an employee is conducting personal business on the employer's premises, and the employer has not expressly granted the employee privacy rights, by way of the employer's policy manual/handbook, or some other express written contract, then a co-worker can surveil another co-workder, and the only recourse is to ask management to prohibit it -- or, quit your job.
That said, there are laws against harrassment based upon discrimination (race, color, religion, nationality, gender, disability, age -- others may be applicable by jurisdiction). Also, there are laws in some jurisdictions (e.g., NY "evesdropping law") which can sometimes be brought to bear on the sort of situation that you complain about, depending upon the precise facts involved.
But, as a general rule, if an employee wants privacy in his/her communications, the only way to get it is to leave the employer's premises entirely.
If the facts were as you describe, there would likely be a number of potential legal claims that could be made, e.g., torts of public disclosure of private facts, intrusion into seclusion, false light attribution, defamation of character, trespass, interference with contract, negligent supervision -- as well as a claim under Texas Statutes Code of Civil Procedure Chapter 123, concerning interception of communications, and under the Penal Code Chapter 16.
There is no independent action for workplace harrassment in Texas. Harrassment claims must be in retailiation for unlawful discrimination under federal civil rights laws. This is a typical misunderstanding of laypersons. Harrassment claims are typically tied to sex discrimination, e.g., a supervisor who makes intimate advancements to a subordinate, and after being rebuffed, goes on a campaign to harrass the subordinate into either quiting or submitting. Similar actions could be tied to race, color, religion, nationality, age, disability.
An employee who constantly harrasses another employee, if it could be demonstrated that the employee's goal was to get the employee terminated, could be sued for interference with contract, because the harrassment is for a wrongful purpose and intended to terminate the employment contract between employer and employee.
But, this would be a private legal action, and you would have to bear your own legal fees, whether you win or lose -- meaning that you would have to have rather large damages in order to overcome the cost of litigation. And, that's why such suits almost never materialize -- plaintiff's can't afford it.
You may want to consider that the harrassment you have suffered may actually be based on discrimination. Sometimes discrimination can be alleged so as to get to the underlying retaliation. But, unearthing the totality of the circumstances is beyond the scope of this forum. You need to get a consultation with a local employment attorney who can discuss all of the facts in detail -- and in private.
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