English-style nicknames and pet-names commonly use shorter, cropped names.
While Irish-Gaelic traditionally uses a lengthened form of the name to indicate familiarity (though there are a few exceptions).
Seán is a very short name indeed. Derived from the French name Jean and equal to John in English, it is impossible to cut the name any shorter. But there is only one semi-common nickname in Gaelic: Seánán. Seánán is pronounced [SHAWN-awn] and sometimes used as a given-name in own right.
Other, mostly obsolete Irish-Gaelic nicknames are Seac (pronounced [SHAK]) and Seóc ([SHOWK]). They're equivical to the English names Jack & Jock (interchangeably) and sometimes the French name Jacques.
On a related subject, the Scottish-Gaels really excel at the art of nicknaming due, I assume, to their strong influence by English naming culture. There are three forms of John in Scotish-Gaelic: Iain, Eòin, and Seon. Seon is obviously the equivalent of Irish Seán. I mention this because the nicknames of Seon are many:
Seoc- Jack, Jock Seocan- Jackie, Jacky, Jenkin Seonaidh- Johnny Seonaodh- Johnny Deoc- (a varient of Seoc) Jack, Jock Deocan- (a varient of Seocan) Jackie, Jacky, Jenkin
Now when Seán is used in English, you know, without the accent, the only nickname I've heard is Seanny (pronounced [SHAWN-ee]). It's kinda goofy, but pretty common for my friends when we were growing up in the American northeast.