I have a basic but serviceable waffle iron I bought at a charity shop for a fiver. If I don't mind a waffle pattern on my sandwich, is there any real need for a Panini press?
Maybe! There's one big potential issue: panini presses are hinged differently. There's a bit of variation exactly how, but the effect is that the entire top can lift up, so that it can press down flat on whatever you stick in there. Waffle irons are just designed to be filled, so they hinge at the back. If your sandwich is much smaller than the waffle iron, you can put it in the front part (as far from the hinge as possible), and it won't matter too much, but if it takes up a decent fraction of the area, it'll be squeezed pretty tight at the back compared to the front. This will probably make it cook unevenly, and depending on what you've got in there, might squeeze things out.
Some panini presses also have better temperature controls than waffle irons, but that's probably not as much of a concern.
So I'd give it a shot and see how it works, plan on possibly flipping your sandwich around halfway through, and don't be too upset if it's not pretty!
(rfusca's comment above is also good - if you're able to take off the hinge, you'd avoid the whole problem.)