Sure happy to help...
One caveat... various shipping entities frequently change their rules so it's best to check with the company a get the latest dope direct from them.
Ok then as a NON-Licensee you are basically down to UPS or FEDEX and I'm pretty sure both require overnight express for handgun shipments. This gets pretty pricey fast. USPS will only ship handguns between FFL licensees.
I'll briefly describe the last couple times I shipped a gun as a non-licensee:
About 8 years ago I traded off a Freedom Arms revolver and shipped it via UPS. At the time UPS was not accepting sealed packages so I had to open the parcel at the UPS office then re-seal it. Additionally, I had to show proof the addressee was an FFL or authorized recipient. the cost was near $70 for next day delivery.
Last Year I traded off my melted 4516. I got on the USPS website, prepared and payed for the shipping via priority mail with my go to transfer FFL guy's address as the shipper. I had arranged in advance for both dealers to exchange FFLs via email so all I had to do was carry the gun and open package with pre-printed, pre-payed shipping label over to my guy, he enters it in his A&D book sealed the box and hands it to the postman. USPS small flat rate box was around $6.75 then plus insurance, my bill was around $12 as I recall.
If you have a good relationship with your local FFL that is probably the most cost effective method to ship a pistol to me.
You arrange ahead of time for me and your guy to exchange FFLs and hand him the gun. He enters it as an acquisition then a disposition to me and ships it. I get the gun and enter it as a acquisition, do the work, enter a disposition to your dealer and ship it back. Your dealer gets it, acquisition then disposition to you, the down side is you'll have to fill out another 4473 pros: less costly. cons: more paperwork and if you're not pals with your FFL... possible multiple transfer fees.
You can ship direct to me though, just not through USPS. United Parcel and FEDEX both have sections on their websites outlining procedures for shipping firearms by individuals. Best to check with your chosen shipper direct to get the latest rules & procedures. Which one is preferred? Ehh, 6 to 1 to half a dozen of the other

No difference IMO, I go with who's closer. For me UPS is less than a half mile away while FEDEX is 70 miles.
DO NOT make, or leave any markings on the box that indicate it contains a firearm. (this includes business names that include "GUN" in the title) best to address it in the name of the licensee.
DO Insure the shipment.
DO NOT ship anything to me unless I have given you the go-ahead.
NOTE: I can only return ship firearms to an official address as noted on your DL. I will need you to text or email me an image of your DL, CCW, municiple water bill, or some other official document with your name on it and the address to where the firearm is disposed to when I'm done with the work. Not my rules but ATF's so don't shoot the messenger
For example. You send a gun to me for whatever... You live in North Carolina. While I have your gun on the bench you go on a trip to Alaska and and want the gun shipped to you at your vacation spot in Sitka. Sorry no can do... I can only ship it to the address on your DL , you know, the DL pic you texted to me before shipping the gun to me back before you left for Alaska, yeah that address

Again, not my rules, but rules I am bound by.
Now, on my end... I receive a firearm from you. It gets entered in my book as an acquisition. I then perform the work. Upon completion I will prepare the shipment via USPS Priority Mail and enter the gun's disposition in my book. I will request a pickup and when my Postman stops by I'll hand him the box containing your gun, inform him it contains a firearm and ask him to scan it in right then. I will have either emailed or PM'd you the tracking number.
And take note: When I return ship it's going to be marked 'delivery confirmation required' AND 'adult signature required'. If nobody is there to sign for it hopefully your carrier leaves you a notice and takes it back to the shop instead of dropping it on your porch
If you know in advance no one is going to be home to sign for it you can let me know to mark it 'hold for pickup' in which case you'll have to go fetch it at your post office.
I think it best NOT to have a gun sitting in the mailbox or laying on the porch all day until someone gets home, especially considering all the parcel stealing we've seen of late.
That's about all I can think of right now, I hope this clears things up somewhat
Cheers
Bill