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, and 24VAC30-73-40 sends entrance permit applications to the designee "for the county in which the work is to be performed." VDOT's own permits page says land use permits "are typically obtained at the VDOT district or residency office serving that locality." The field review looks at the entrance and decides whether a pipe has to go under it, and the permit then names the pipe's size and type 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 pipe ends "shall minimize any hazard the pipe ends or structures may present to an errant vehicle."
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, except for the duties the regulation gives VDOT, and it lists those by type of road. 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. The rest of the list covers curb, gutter and sidewalk, including a road with shoulders, ditches and a sidewalk belonging to VDOT, where VDOT maintains the portion of the entrance that extends to the back of the sidewalk.
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. On a shoulder and ditch road with no VDOT sidewalk, everything from the shoulder back toward the house, concrete included, is the owner's to maintain, apart from the pipe work the regulation gives VDOT.
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.
Drainage is among the reasons the rule names for pulling a residential connection back in, at the option of the Department: where drainage installation costs are excessive, or where drainage complications are obvious. So on the North Carolina side, drainage alone can bring a plain house driveway back under the permit requirement, and whether it does is the Department's call. 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, show 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 permit is the owner's to obtain: 24VAC30-73-90(B) says "The property owner shall obtain an entrance permit," and on a shoulder and ditch road it makes the owner responsible for installing the entrance as well. Settle in writing who buys the pipe and who sets it before either one happens.
The order that keeps a pour on schedule is not complicated. Get the review, and the permit with the pipe named on it. Put in a locate request before anyone digs or grades, and the answer on calling 811 ahead of a pour covers the timing. Do the clearing and grading the permit calls for, since the LUP-PE wants all clearing and grading for sight distance done "prior to using the entrance for construction purposes," and a concrete truck on the entrance is exactly that. Then set the pipe, build the subgrade over it, and 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.