With the capabilities of most EHR systems, the system itself can usually do the actual verification itself. My personal opinion is that if ANYTHING is off, that should be reviewed by a biller. Unless your front desk are very experienced and detail oriented, they usually do not have enough training or experience to understand how much impact what seems to them like a minor thing can result in denied claim and the headache that causes down the road.
In my healthcare organization, in most practices the entire responsibility falls on the front desk. In my particular practice (at my insistence), the front desk is responsible to click the button to verify, and unless everything is a match, that falls to a worklist that my biller reviews.
If for some reason, you have humans (front desk or billers) doing a lot of actual verifications (website or phone), you may want to check with your vendor about additional capabilities.
Regarding how often a verification is done, my personal recommendation is at least every 60 days. I've worked places with timeframes of only good for 7 days, to only the first time each calendar year.