Hi there!
Former BCG Consultant here. What do you mean by "collaboration with BCG" on a daily basis?
I assume you might mean between client and BCG consultants - but let me know if you mean something else, as the response would be a different one.
Assuming the above: it depends on the scope of the project and the client. There is relatively limited day to day collaboration on a DD, somewhat mid-collaboration on a strategy case and quite intense collaboration on a PMO-type of project. I have seen in the latter cases mixed teams, especially if the client is interested in gaining that BCG knowledge.
Happy to provide more concrete examples on how that works in practice if you are interested.
Hope this helps,
Ariadna