Well, it isn't something you want to be doing manually. The "greasemonkey script" was described as a way to automate this verification.
The thing is that since bitZino doesn't know who is verifying and who isn't. But if even one single verification I do fails, that would be inexcusable and would be not only the last hand I'ld play would be something bitZino would have to come up with a darn good explanation as to how it happened.
So while bitZino can be probably fair on any hand, most hands are played without anyone doing the verifying. Unless you are recording the info in each hand (and how you played), you can't go back to previous hands to verify. Again, a greasemonkey script can do all this but the bitZino web app doesn't.
With BitLotto and SatoshiDICE for example, proof of fairness can be determined long afterward, as nearly all the info is in the blockchain. For SatoshiDICE, their hash file plus their secrets file is also needed. Their hash file is a static file that has been archived independently, and the secrets file is appended to daily and can be verified at any point in time using the hash file. The additional info necessary for BitLotto comes from the results of a state-run lottery (Megaball) which occurs after the monthly draw deadline.
If bitZino (or someone else) were to create a mobile app version that did the verification on each hand, then verification for each hand could be performed, in real time. Of course, that should be an open source app so that the method can be inspected to ensure it is doing the verification properly.