I have never actually seen a CSM touring. Usually, that is the (P|D|T)SM (production, deputy or touring) stage manager.
Usually a show will have a rider which is distributed to all the venues being used which says how much of this, how much of that, what water is need, what specifications the venue needs to meet (including power, internet, phone) - you may have heard one of the most famous rider demand - "A bowl of M&M's, with the brown ones removed" - Van Halens show was apparently so technical demanding in it's day that that clause was placed as article #126 to make sure the rider was obeyed. The rider is usually written up by the company stage manager or the technical director. Then as touring SM, you basically need to ensure the rider is met, and raise hell if it isn't.
Basically, the TSM should be able to walk into the venue, plop the paperwork on the prompt desk and direct the bump-in without having to worry that there are only 120 channels of dimming instead of the 132 requested, that the live internet feed should be a matter of jacking into the network because they have already set all of that up.
This is the sort of heirachy we have in australia anyway, I doubt it will be too different, but the company manager (big boss) resides in the company, they manage the big picture. You, the (T|P|D)SM manage the single show. You need to set up your call board, in collaboration with the CSM (to make sure all times are correct etc). When it comes to the rider being written, you and the technical director tell the CSM exactly what you require, they distribute it to the venue, organise all that junk, and before tour present you with an exception report which may say things like
"Venue A:
*Only has 4 dressing rooms
*No Foyer
*Will have tech guy there to assist with networking
Venue B:
....
"
However, you should never see something like
"Venue C:
*Need to hire another 12 channels of dimming"
It is the CSM's job to instead have
"Venue C:
*Need to pick up 12 channel dimmer from xyz, 1231 this street, paid in full"
That said, all companies are different and you need to talk to the people you are working with the find out where your jurisdiction ends, and theirs begins.