I worked at a law firm and I wanted to build a National ERISA practice. Since the partners there weren’t interested in sharing clients, I tried my best to get some. I drafted an email to an advisor I knew from the third party administrator (TPA) I left several years earlier. To say something about my work ethic, they eventually needed to hire 3 people to replace me, so their legal fees went up. I mentioned that my fees were much more competitive than the TPA’s legal fees and I could offer an attorney-client relationship. The TPA found out about my email and contacted the managing Attorney who didn’t like me from Day One. The tongue lashing is memorialized in my Kindle Book about 20 of you actually bought, but let’s just say that everyone takes a beating someday (obligatory GoodFellas reference).
As far as employees are concerned, you should be concerned with their social media posts as long as they identify themselves as your employee. In my area, I’ve seen school district employee say things on Facebook that no school district employees should be saying as school district employees. There is nothing wrong with employees expressing themselves as long as they’re not expressing themselves while identifying themselves as your employee because you may not want to be linked to their views.
You have enough things on your plate without worrying what an employee may post as your employees. You don’t want to be saddled with views and actions that may not reflect well on your organization. This isn’t a matter of free speech (the First Amendment is protection from government interference), it’s a matter of you having fewer headaches.