Employers have valid concerns about the way their employees conduct their personal lives, such as breaches of commercial confidentiality or bringing the organisation into disrepute. Such activities could count as gross misconduct, and even justify dismissal. This is the same whether the misconduct happens online or offline.
However, employees also have a right to a personal private life, and provided they do not breach reasonable conduct guidelines, employers should respect this. They wouldn't follow an employee down the pub to check on what they said to their friends about their day at work. Just because they can do something like this online, doesn't mean they should.
A responsible way to handle this is for employers to negotiate a reasonable conduct policy with staff, and make it clear what is expected of them in their private lives, both offline and online. This way staff will be less likely to get an unpleasant surprise when they find out their employer doesn't approve of something they said or did. If there is no conduct policy in place already, concerned employees should talk to their union if they have one, to discuss working out a conduct policy in consultation with the workforce.