I think the debate is partly obscured because Pollock was supposed to be "destroying the image" and whatnot. I've had many people tell me you need to see the complexity of the paint in person. I've had many people tell me the meaning is in the action of Pollock in painting it, not the paint. I've had many people tell me the real story is behind Pollock himself and how he came to paint the pieces. Pollock himself said he wanted the general public to treat the works like music to go into their unconscious and go "oh, that's nice".
And of course, when pushed about what they actually are experiencing I've gotten hogwash like this:
Linky.
I'd say "random" is an inaccurate word, because one of the few things we can determine is that he aimed for equal paint distribution. The kind of "pseudo-random" people draw if you ask them to draw "random" (dots evenly distributed vs. dots in clumps). The other obvious hints of intelligence are that the paint doesn't merge and mix into brown muck, the color choices themselves, and the drips themselves, on some works more than others.
However I will say I have never had any of these things transmit anything to me other than "this is paint".
If people are gleaning any meaning, it would be interesting to put them in a room and see if there is any consensus. That is, of course, if they can put the revelations into mere human tongue.