If I've understood this correctly, you've got
a) patient sees a Doctor#1/a specialist (?)
b) Doctor #1/specialist does what doctors do and discusses a case with another doctor who they have a working relationship with/know is a good doctor and could be in a position to accept a referral.
It's a convention between specialists in particular, that they discuss first - to just refer with no warning/notice is regarded as extremely rude. (specialist to specialist referrals are somewhat different to the bog standard NHS GP-Hospital referral system, as are second opinions).
c) Doctor #2 when aware that the patient is actually a friend, being cognisant of the possible conflicts inherent in seeing a friend in a professional capacity, says 'I can have a chat to find out the most likely suitable path, but I'd need to refer them on to a colleague because of our personal connection and because my colleague may be more experienced/better at what the patient needs. The chat would enable me to identify which of my colleagues would be the appropriate specialist whom I trust most with a friend '.
d) Doctor #2 decides to contact friend and inform her of the situation, assuming that friend had known of the pre-referral discussion. Friend is then shocked because it's all a bit of a surprise.
e) Friend decides whether to see Doctor #2 with a view to being referred (and seen quickly, as favours tend to have that side effect) or not.
Or you have Doctor #1 knows you are friends, but Doctor #2 is still the best person for trusted and knowledgeable advice. #1 therefore speaks to them privately and discloses 'this is Patient, I know you know her as well, are you able to advise on the appropriate course anyway?' Doctor #2 says 'Of course, I can have a chat and then see who I need to talk to so that Patient can be helped best. Informally, as I don't want the referral myself because she's somebody I know and that can get a bit weird'
It's not the same as a gossip over the garden fence about 'You'll never guess who I had in today complaining about her piles!'. It's what doctors do; discuss patients/cases with a view to getting the most appropriate advice and knowledge for the patient.