Does a freelance graphic designer send a brand identity project to a printing company or is that the clients job?
I've done it both ways.
Sometimes a client of mine will already have a relationship with a printer and want to handle the printing themselves. In this case I make sure the the print ready files are done to the best of my abilities (I usually try to find out what printer they are using to best finalize my files for that printer) and either send the print files to the client or to their printer directly.
Other times I have clients who would like me to handle the printing for them. They either don't have the time or the knowledge to handle the printing. I prefer this way because it both makes me more money, and I have greater control over the final result (usually because I am working with printers I know and trust) and can better ensure the final product will be right.
That is a decision between the designer and the client to make. There's no "right way" to handle this although many designers prefer to do the hand-off as they're more likely to have the vocabulary and domain knowledge to communicate clearly with the printer about what is needed.