Wowhead already has links you can copy and paste in-game to load a transmog set.
For example:
http://www.wowhead.com/transmog-set=...legear-recolor
Click "Links" at the top, then copy the "Ingame Link" option, and you get this (I added spaces for forum readability):
/run local function f(i,b) DressUpItemLink("item:"..i.."::::::::::::9:"..b) end f(113636,566) f(113649,566) f(115540,566) f(115541,566) f(115542,566) f(115543,566) f(115544,566)
If you paste this in-game, the dress-up window will open with all the items, and you can save it as an outfit in your wardrobe.
All an addon would need to do would be:
(1) Generate a data string. It wouldn't need to include any functions, just the item data, so for the above example, it would probably look something like this: "Living Wood Battlegear (Recolor),113636,566,113649,566,115540,566,115541,566,115542,566,115543,566,115544,566"
(2) Send it to the specified player via the SendAddonMessage API.
On the other end, the addon would need to:
(1) Listen for addon messages with the appropriate prefix.
(2) Split the data string up into its parts and load the items into the dress-up frame, with an option for the receiving player to save the outfit using the provided name (or a custom name).
There would be no need for the addon to handle any actual transmog stuff, since the default transmog UI already lets you transmog an entire outfit in one go. However, you could highlight recently received sets in the list (with a star icon or something) if you wanted.
I don't have either the time or the interest level to actually write such an addon, but it wouldn't be very hard to do.