It is easy to visually check the brightness variation and chromaticity variation of an LCD monitor, but first you'll need to know exactly what they are before you can look for them.
Brightness variation scatters brightness around the screen while chromaticity variation scatters colour around the screen. This isn't as easy to notice as brightness variation, but it makes graphics-related displays unnatural and causes deterioration in colour reproducibility.
Every LCD has some degree of brightness and chromaticity variation, but there are many products where the variations become more obvious when the brightness is lowered.
Brightness and chromaticity variation can be checked with standard Windows or Mac OS X functions. All you need to do is to set the desktop background to "Monochrome" and look at the whole screen after taking a few steps back from it. You just need to check for uniformity and to see if any of what's mentioned above is visible. It's also important to cycle through different colours so your check is more thorough. On the EIZO website it is suggested that you change the desktop background to black, white, gray, then 100% red, green, blue, cyan, magenta, yellow, and then any "near-white pale colour".
Generally speaking, the center of an LCD screen is the brightest and it gradually gets darker towards the edges and there's nothing to be concerned about if there is not a big difference in brightness between the central and peripheral areas. However, some products may show a more obvious difference than others.
Also, you may be surprised to find more variation than you expected when gray or a near-white pale colour is displayed.
A monitor which performed well in this particular test is the EIZO FlexScan SX2462W and more information about this device, along with details on all of EIZO's products can be found on the EIZO