Praise be to Allaah.
We put this question to Shaykh Muhammad ibn Saalih al-‘Uthaymeen, may Allaah preserve him, who answered that this is bribery and is a path to corruption. It could lead to these doctors deceiving the patients and lying to them, and transferring them from the government hospital, which is free, to the private hospital on the grounds that certain treatment or equipment is not available in the government hospital when in fact they are available there. And Allaah knows best.
Sheikh Muhammed Salih Al-Munajjid