A private individual can send a firearm to an FFL in their state or another state without going through a dealer.
The shipment can be for gunsmithing services; repair, refinishing, modification, warranty work etc. in which case the receiving FFL can ship the gun directly back to the private individual.
An individual can sell a gun to a dealer in their state or another state and ship direct to the FFL.
An individual can send a gun through a dealer to transfer to a person in another state. the receiving dealer would transfer the firearm in accordance with standard NIOCS protocol.
What can you ship:
The individual can long guns to a dealer/gunsmith (anyone with FFL) through Post Office, FEDEX, UPS etc.
The individual can only ship handguns to a dealer through UPS, FEDEX or other common carrier. Only dealer to dealer can ship handguns in the mail.