You kind of have to assume that the hardware isn't lying when it tells you that it's properly copying data between the two drives. The "healthy" status just means that both drives are running, both drives have the formatting that says they're part of the RAID array, and they're both responding.
If one drive were to have some sort of write error (there are lots of checks in the process to make sure the drive gets the exact data coming from memory all the way to the drive platter), it would signal the controller just like any single drive should do, and the drive should be dropped from the array. So as long as the array shows Healthy, there is assumed to be no problem.
I don't think any 3rd party software can scan each drive and compare it to the other, except maybe by pulling them out of the system and using some other software to do a bit for bit scan. Since reading the drives simultaneously is a function of the specific driver for the controller, no other software is likely to be able to hook into it properly and take direct control like that. Some drive controllers might have a "verify array" option, either in the boot setup or in the software monitor, that would actually do a comparison between the drives. Not sure how it would determine which drive had the correct data, if they did actually differ.
Just like with any drive though, it could be possible that a drive has a bit that gets changed by cosmic rays after it's written or has a bad sector, which wouldn't be detected until later. I'm not sure how RAID1 deals with differences between the two drives when it reads a bit of data, but since current controllers provided improved read speed by reading alternating stripes from each drive (i.e., it doesn't read the same bit from both drives at the same time), it may not even notice it.
