What the pipe is there to do
Where the road in front of a house is a shoulder and ditch section rather than curb and gutter, the ditch is the road's drainage, and it has to keep running past your frontage whether you build a drive or not. A driveway crossing it is fill sitting in the ditch line, and fill in a ditch is a dam unless something carries the water through it. That something is a culvert pipe.
The rule behind it is short. 24VAC30-73-130 says entrances shall be constructed so as not to impair drainage within the right of way, and so that surface water shall drain from the highway. Nothing in that sentence is about your driveway. It is about the road's water still getting where it was going. The pipe is how a solid entrance and a working ditch share the same ground.
The ditch decides, and somebody reads it in the field
So the honest answer is that it depends on what runs across your frontage, and the person who reads that is not sitting at a desk. On a VDOT maintained road the LUP-PE puts it on a field review by the district administrator's designee, which VDOT's permits page describes as the district or residency office serving your locality. That office looks at the entrance, decides whether a pipe has to go under it, and writes the pipe's size and type onto the permit for the permittee to furnish. The size arrives on the permit, set at that review.
That has a practical edge to it. Any pipe size quoted to you before the field review is a guess, ours included, and the pipe gets bought after the review rather than before. The entrance permit walkthrough covers the paperwork side of this, from staking the centerline to the fee. This page is about the ditch.
Who sets the pipe, and who gets billed for it
24VAC30-73-90(B) puts the work on the owner. On a shoulder and ditch section road the property owner is the one responsible for installing the private entrance in accordance with VDOT policies and engineering standards, and the regulation separately makes the owner responsible for all grading beyond the shoulder. The owner may instead ask VDOT to stabilize the shoulder and install the entrance pipe, and VDOT may then install the pipe at the owner's expense.
If VDOT does install it, read the rest of that passage carefully. VDOT provides a cost estimate for the installation, and then bills the property owner the actual cost of installation. An estimate and a bill are two different numbers, and the regulation says so out loud. No rate is published, so ask the residency what they expect before deciding who sets the pipe. One more detail from 24VAC30-73-130: pipe ends are reviewed on their own by the district administrator's designee, and grading or treatment at the ends has to limit the hazard those ends present to a vehicle that has left the road.
Ten years from now, who cleans it out
Worth knowing before you own one. Under 24VAC30-73-90(E) the maintenance of a private entrance is the owner's, with two exceptions written into the regulation. On shoulder section highways VDOT maintains the portion of the entrance inside the normal shoulder. On highways with ditches VDOT cleans the drainage pipe at the entrance, and may replace it where that is necessary to preserve the highway and protect the traveling public.
That is a narrower promise than it first reads, and it is still useful. The pipe under your entrance sits in VDOT's own drainage, so VDOT has its own reason to keep it open. Everything from the shoulder back toward the house, concrete included, is yours.
Across the line, the ditch is what pulls you back in
North Carolina starts from the opposite assumption. 19A NCAC 02B .0601(a) opens by requiring a permit for any driveway or other connection built inside a state system right of way, then, in the same subsection, treats a residential connection as the usual exception. The rule's own word for that is normally, and it is the word carrying the weight.
Two of the four conditions the rule names for pulling a residential connection back in are about water: where drainage installation costs are excessive, or where drainage complications are obvious. So on the North Carolina side, half of the named reasons a plain house driveway can be pulled back in are about the ditch, and the District Engineer decides whether either one applies. NCDOT's own Policy on Street and Driveway Access encourages the property owner to contact the District Engineer to identify safety and design issues and to arrange for installation of driveway pipe. Worth doing before anything gets graded, and the North Carolina driveway rule is written out in full on its own page.
How a ditch line changes the estimate walk
Tell us there is a ditch when you call. On the walk we look at where the water comes from and where it goes once it is past you, because that changes the shape of the drive down at the road and it changes which way the surface has to fall once it climbs onto your ground. The concrete is ours, apron included. The pipe and the permit sit with the owner, and the free written estimate says that in plain words rather than leaving it to be discovered later.
The order that keeps a pour on schedule is not complicated. Get the review, get the pipe named, set the pipe, build the subgrade over it, then book concrete. Concrete last. A drive poured across a ditch line that was never piped is a drive the first hard rain goes to work on, and that repair can mean a tear out rather than a patch.