Employee Verification

I'm searching for a new job and have had some promising interviews. Most recently, a very promising position that would be a great fit has come up and I've been invited back for a second interview for a nationwide company. I don't want my current employer to know that I'm looking for a job elsewhere as I'm worried if I can't find a new job it will get me on the shitlist.

My question is - how blatant is the employment verification process generally? Is it something that only happens after an offer is made, or is it more following up on provided references?

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