Recently, I got a pair of Elliot Lucas PW216-6" engineer's pliers from the UK for around $12 shipped.  The seller claimed, "like new", but they obviously weren't, and the center pin/rivet that holds the two halves together is quite loose, allowing around 1/16 inch of side-to-side play, an issue that's become quite annoying in the week I've been carrying them.  Spear and Jackson, Lucas's current parent, doesn't mention any kind of warranty, and even if there was one, as an American, I'm probably not covered since we're not one of their markets.

So, how do I carefully go about tightening them up?

If it is a rivet, I would try an arbor press first.

I have to assume that the center pivot is firmly fixed to one side of the pliers?  Take a punch or metal rod that is just less than the diameter of the pivot and start tapping the loose side with a hammer.  A lot of tapping - not smashing - may get the pivot pin to bulge a bit and take up the slop. 
You might consider heating the pivot pin til it just turns the slightest bit red, and tapping it lightly.

You will want them to be laying flat on an anvil or chunk of heavy plate.