Definitely drill through: easier to clean, easier to remove the dog by popping it up from below or down from above, and you can push the dog down to flush or nearly flush as necessary. Remember to clamp on a backer block when you drill through.
I'm planning a bench top. I'm planning the thickness of the top and
> the spacing of the bench dog holes, but I have realized that I don't
> know whether the dog holes should be drilled completely through the
> top, or stopped at a specific depth. Instinct tells me to drill
> through, because cleaning the holes would be tough.
>
> I'll be using Lee Valley brass dogs or something similar (possibly even
> dowel segments, given my budget.
>
> TIA for any advice.
>
> Kevin