The engineer is responsible for designing per code. So, if you are on the contractor side, you would not be responsible for code conflicts. Past that, "responsibility" is a nebulous term which can vary widely, based upon the type of project involved and how the specifications are written.
AIA guidelines are just that. The drawings and specifications are the contract documents which are binding for the project.
If the engineer is providing installation drawings, then he is responsible for designing a system that coordinates with all other trades. Generally, the engineer does all that, but since he is not providing installation drawings, there might be some issues.
In my experience, the engineer is not providing installation drawings but a "general design intent" which may be very exact or may be very general.
In the specs I use, there are clauses that the intalling contractors are responsible for specific installation layout and coordination with other trades.
The installing contractors are responsible for providing coordination drawings which are typically signed off by all the trades indicating pipe/duct/conduit elevations and distances from column lines.
The engineer typically reviews them, makes comments, but may not have to approve them.
If something doesn't fit or cannot be shifted a few feet to miss structure or other trades, then the engineer is typically brought in to assist the coordination.