Hi, as a private patient, I used the HPA Saude hospital at Alvor for a consultation and a very minor op. I guess it depends what type of op you need, mine was just a skin cancer thing. I presume not all hospitals do all ops. I don't have Pt health insurance so just paid, it wasn’t too expensive although I notice their prices have increased quite a lot in the last couple of years.
I find them helpful, pleasant and virtually everyone spoke English, they were all incredibly multi-lingual. I go there for regular check ups now.
Only complaint about them is that they seem to have the slowest computer system in Portugal, which is saying something. You might find you lose the will to live while they wait and wait and wait for their computer to do something, anything at all, such as find your booking.HPA really need to invest in a new system.