I haven't noticed this as much as my correspondent, actually, but that probably just reflects the kind of messages I get. There's the occasional use of ellipsis dots, certainly, but then there always was on the Internet (especially in chatroooms) and in informal writing the practice goes back centuries. There are many literary antecedents. In recent times, Harold Pinter was one of the masters of ellipses as an indication of an unfinished thought or an unstated implication. Take a look at The Caretaker. Here's Aston talking in Act Two:
'They weren't hallucinations, they ... I used to get the feeling I could see things ... very clearly ... everything ... was so clear ... everything used ... everything used to get very quiet ... everything got very quite ...'
And Davies ends the play on an ellipsis:
'Listen ... if I ... got down ... if I was to ... get my papers ... would you ... would you let ... would you ... if I got down ... and got my ...
Long silence. Curtain.'
All this before texting was ever invented.
I wouldn't call it laziness. There's certainly an element of convenience behind a typing usage, as a period is often easier to type than, for example, a question-mark (which may involve a shift key). But the usage is much more than that. It's a further example of the way informal expression on the internet is getting closer to what happens in speech.
Imagine I'm telling you, face-to-face, what I've just written. You'd hear the pauses, the continuative intonation patterns, the variations in tempo (allegro, lento) which would show me actively processing what I want to say. Prosody is the main way of showing people 'thinking on their feet'.
Writing displays none of this. You will never know what pauses I had between the various bits of a paragraph. In fact, if I recall correctly, the paragraph before last went something like this: 'I wouldn't call it laziness ... There's certainly an element of ... convenience behind a ... typing usage, as a period is often much easier to type than ... for example ... a question-mark (which may involve a shift key) ... But the usage is much more than that ... It's a further example of the way ... informal expression on the internet is getting closer ... to what happens in speech ... '
This is becoming Pinteresque. The ellipses reflect the thought process, the decision points, the places where I was thinking how exactly to put what I wanted to say. (In Pinter they usually have a more menacing purpose.) In speech, these decision points are there for everyone to hear. And if (unconsciously) you want your writing to reflect speech, ellipsis dots are an easy way to show it.
They also show that punctuation isn't as important as people sometimes claim it is. I know we all have to use standard English punctuation in our formal writing (and Gove help us if we don't!), but the informality of Internet expression shows that these are conventions of correctness that bear little relationship to clarity and ambiguity. There was little by way of punctuation in the earliest English writing, in Anglo-Saxon times, and the texts come across just fine. Indeed, one can dispense with all punctuation and still get one's meaning across, as has often been shown - though, because we're not used to such things, it does become more difficult to read. Take the present paragraph, for example:
'They also show that punctuation isnt as important as people sometimes claim it is I know we all have to use standard English punctuation in our formal writing and Gove help us if we dont but the informality of Internet expression shows that these are conventions of correctness that bear little relationship to clarity and ambiguity indeed one can dispense with all punctuation and still get one's meaning across ...'
It's like the end of Joyce's Ulysses. The main effect, in such rewriting, is phonetic, not semantic. We miss the guidance punctuation provides about where to pause and take a breath and thus (cognitively) to assimilate what is being said. That's why punctuation developed in the first place: to help people read aloud easily. Ellipses, I suspect, are there chiefly for phonetic reasons too, not semantic ones.
Given that character totals are at a premium in texting and Twitter, you might think it surprising that there are ellipses at all - often many more than three dots, with people just holding the period ley down for as long as they like ............ A lot of dots reduces your character count. This suggests that users see a real point (sorry) in using them. It probably doesn't affect their textingtweeting style too much, as most texts and tweets don't use the full 160/140 characters, so there's still plenty of opportunity to put in some extra periods. And maybe that's another factor: users know they have room to spare, in routine messages, so they let their periods roam. I bet there aren't so many when the content gets more complex and structured, as in ads or news announcements.