I'd expect my private company to expect a sick note (or nowadays a fitness for work note) detailing from a doctor that in the doctors expert medical opinion the employee needs to be signed off for whatever reason. Once they have that note I'd expect them to either trust that doctors note, or if they have doubts (and as most contracts explicitly allow) request a second opinion paid for by the company. That is the correct process to investigating suspect sickness leave. Jumping to intrusive (and presumably expensive) surveillance within 4 weeks certainly isn't the correct appoach. More advanced employees may would instead spend that money on looking at what it is in their processes and workplace that is causing the stress and/or providing counselling/stress management resources to the employee to help them cope.
Certainly I strongly doubt public and private companies differ greatly in how they handle stress and other long term sickness - though from personal experiences I know the stress placed on public employees is often much higher and there seems to be (on average, etc etc) less capable managers.