As someone born in the late 80s I've only heard about it from the press. I really sometimes omit periods in one-sentence text messages, but sometimes I don't, and it wouldn't come to my mind to ascribe any semantics to this.
My personal nitpick is people writing a stream of one-sentence messages without punctuation instead of a longer structured message - that definitely carries the connotation of unstructured thinking to me - but it doesn't have much to do with full stops per se.