I say if that monitor is 1680x1050 512MB is adequate. If you really want to strive for high performance and multiple graphically intensive applications at once, 1GB is a good option. It all comes down to how much you are willing to pay.
If it's 4870/260gtx or above, 1gb is better. The gpu is powerful enough to actually make use of the higher ram (running at playable fps at large enough resolution or heavy enough future rendering engine). It's not only about resolution. Future, heavier games will take more ram and gpu power to run at any given resolution as older games. The max playable resolution on the same card will shrink as time goes on. What's overkill today will be useful tomorrow.
When you look back at the amount of time between each of 64->128->256->512mb starting to be useful, the 1gb switchover is right about now.
512MB should be fine for 1680x1050, but the 1GB memory would be a good choice for 1920x1200 or above. Only the GTX 260/280 and the HD 4870 have the memory bandwidth for it though. The 4870s high memory clock speeds make up for its smaller memory bus compared with the GTX 260 and 280. The HD 4850 still uses GDDR3, so can't really take advantage of 1GB of ram with its lower memory bandwidth.
I know the 512 is enough to power the 1620x1050, but I'm thinking more of the other stuff that will be expected from graphics cards in the future. Its all going to take memory, and I'll still want my 512 for graphics wont I?
The obvious answar is get a 1 gig version of the 4870, but with a DDR5 shortage, it'll probably be a while before It trickles down to my price range, especialy 1 gig worth.
I'll be hard pressed to justify a 1 gig 4850 to the little woman (she is a little pissed that the 7900gs she got me isn't good enough anymore), too much more, and I'd have a hard time justifying it to myself.
I guess the basic questions are: is this coproccessing and other stuff going to happen sooner or later? it looks like sooner to me. and Is gthe 256 bit DDR3 combo good enough to adequitly handle 1 gig. I know it'll be faster than if it has to access system memory.
p.s. I'm also noticing 4850s getting rebates already. seams like thats happening quicker than I thought.
256-bit bus and GDDR3 will not be adequate for 1GB of ram, look at the benchmarks of the 1GB version of the 8800GT/8800GTS, they gained almost no performance at any resolution by adding the memory to them and in some cases lost performance.