People normally have only one identity, which covers them for both work life and personal life. Asking people to totally sacrifice a private life in favour of work is unreasonable.
This isn't such a problem offline, but it becomes a much bigger issue online where everything is written down. It's easy to see why an employer might worry about what staff may be writing online, but this does not give them grounds to tell you to stop using the service in general.
What they can sanction you for is anything you do that they have a reasonable belief could prove detrimental to the company.
If they can prove that your conduct online has caused detriment (or represents a genuine risk to the employer’s reputation, even if no harm has been caused yet), they could take disciplinary action against you and potentially dismiss you as a result. If the harm to reputation or leak of confidential commercial information were severe enough, they could also sue you for compensation.
If your employer is especially concerned about Facebook use, the best advice is never to mention your work on Facebook, and to be very careful who your Facebook Friends are. Most tribunal cases result from a co-worker or manager printing off a paper copy of a Facebook entry they object to and sending it to HR.
The law does not regard social media sites as 'private' spaces, no matter how few in number your Friends are. This means it is unlikely to be a breach of privacy for your employer to look at your Facebook account, unless they have hacked into it.