The compositions - which are coupled with moving emoji designs - last just five seconds and can be sent to friends over the messaging platform for free.
The musician said he had at first considered the project a "strange proposition" but then changed his mind.
"I thought, you know what, why not? Something fun, and nice and new."
A video shows the 73-year-old in the studio creating short tunes for 10 emojis, which capture emotions such as "flirting", "blushing", "love" and "lust".
He uses instruments including the Moog synthesizer, harpsichord and electric guitar, as well as his voice, although the so-called "Love Mojis" contain no lyrics.
"It turned out to be a great laugh at the same time as challenging, as you suddenly realise you've got to compress a musical interpretation of an emotion into less than five seconds," he said in an interview on his website.
"So with some of them, which were five and a half or six seconds, we had to work out a way to take the front off or just squeeze this or just take that bit out."
Some emotions had been harder to capture than others.
"There were a couple that were more challenging; 'flirting' and 'blushing'," he said.
"But I just kind of had a go at it and thought, 'There you go!' And then if I didn't think it was enough, I might just add another instrument to try and get the feeling."
Skype's Mojis, which were launched last September, differ from emoticons in that they can feature sound and video.
Five billion monthly
Head of MIDiA Research Mark Mulligan said the partnership with Sir Paul made sense, because messaging platforms were "the next place digital audiences congregate".
"First there were websites, then social networks and now messaging platforms which number more than five billion monthly active users globally," he told the BBC.
"Skype is trying to reinvent itself in this environment and has clearly considered investing in a 'partnership' with Paul McCartney a useful tactic.
"However, if Skype wants to build the same sort of relevance with younger users that Snapchat and Instagram have, it should really be exploring opportunities with artists two or three generations younger."