I work for a public company in their tax department, and one of the firms we outsource stuff to is Deloitte. In my experience they are very friendly and nice, but also from my experience every firm varies significantly by geographic area. I have worked with all four of the big 4 in some respect, and as with any organization that large, there isn't any real set persona. There certainly tends to be a similar attitude/personality of the team members on each team, but that is largely due to the manager selecting who they want working with them.
An experience I had that kind of demonstrates this, we were working with 2 separate teams from the same big 4 firm, for 2 different projects. The one team was extremely friendly, easy going, and very responsive and engaging when you had questions or concerns. While the other team was very arrogant, dismissive, and condescending when you had questions or concerns. This was two teams from the same firm, but they were out of two different offices in two different cities, point being they vary considerably by location in my experience.
On a kind of side note, one other thing I have noticed that influences how they act, is the personality of the person at your organization whom makes the hiring decision to bring them in. After all the big 4 are really service organizations, and so they tend to be very adaptive to their customers personalities if they are good at their job.
AUD - 2/2013 - Passed!
BEC - 5/2013 - Passed!
FAR - 8/2013 - Passed!
REG - 11/2013 - Passed!
"Do or do not, there is no try." - Yoda