It varies from organisation to organisation.
From a licensing point of view, once you have your MEP rating in your sweaty paw you can fly anything which fits that criterion (unless you need differences training for pressurisation systems etc). But that is only a licence, not a right.
Many places will want you to prove you are not likely to bend or break their precious asset before they will let you loose in it. I forget the exact numbers, but the club at which I instruct want you to do another 5 hours of 'supervised' flying in the same type as that in which you did the rating. Note that this does not have to be with an instructor, just a competent twin-rated pilot in whose capabilities the club has some confidence.
For the Aztec, I think it's something like 10 hours' supervised and a minimum number of multi hours. Not quite sure why they consider the Aztec that difficult as it is slower than the Duchess!
Some places will be happy to take you up for a dual flight and see how you get on before they decide. Others trot out the 'insurance won't allow it' argument.
I suppose it's up to you how confident you are to take a twin out given your own particular experience level. I would suggest that some 'dual' time is of great benefit to you if you are low on twin hours or out of practice.