This depends on the platform, on console you will have to make your part. Publish it, then your pal can work on the rest. With proper communication it's a matter of getting the new version from your friends workshop and vice versa. When you are done only one of you should keep the map uploaded.
You cannot copy and paste customer objects on console.
On PC it's the same process as the console except you don't upload the map till it's ready. Just share the actual file withing your documents folder back and forth.
You can share, copy and paste custom objects as long as you have Steve64b's map editor mod. Simplest way is to have an empty map where you make any assets he needs, send him the map file and he can just highlight, copy and paste.