Depression is a psychiatric disorder, i.e. a disease that needs to be treated.
If any of you feel like you suffer from depression, I'd recommend going to a psychiatrist or psychologist first as either one of these will direct you to the other if needed (i.e. if he notices symptoms of depression). The point of this is to get a proper professional advice on your health.
I can't say for sure about psychologists, but psychiatrists are doctors and thus they are bound by patient-doctor confidentiality, thus whatever you say to a psychiatrist should remain a secret and there's no reason to worry about privacy here.
---
At any rate, from what you've said earlier, I do not see any reason to suspect depression as a cause of your marital issues, though - obviously - if you're worried about it and especially if you're suffering from symptoms like feeling hopeless and sad most of the time, self loathing, lack of energy or constant fatigue, etc. you should go see a doctor.
---
As for counselling, in general I think your issues belong to the couples counselling category and you should look for a specialist in that field.
Personally, I'm not a huge fan of online counselling and I'd strongly recommend therapy that that is conducted in person.