Let's define some terms clearly since there's some overlapping terminology here:

• "Delta.com Account": This is used for logging in to the website. It's a convenience thing -- you don't actually need one to make a booking. When logged in to a Delta.com Account, the logged in person can make a booking involving more than one person.
• "SkyMiles Account": This is where miles accumulate. As mentioned upthread, each individual must have their own SkyMiles Account, and you can't credit your flights to somebody else's account. (Other airlines do allow this, eg: jetBlue Family Pooling.)
• "Passenger Name Record (PNR)": Mostly you don't need to worry about this. This is the reservation file created when you make a booking. If one person makes a booking for two people (whether or not logged in to their Delta.com Account, and whether or not either individual has a SkyMiles Account), both people will end up in the same PNR. This only really matters if something goes wrong (eg: flight gets canceled) as you're more likely to get rebooked together. (Sometimes it matters for sharing elite benefits, but that sounds like a non-issue here.)
• "Ticket": Each passenger gets their own ticket, regardless of how it was purchased. If purchased together, both tickets will be in the same PNR.

So in summary (as already described upthread), each passenger gets their own ticket, and each ticket must have its own SkyMiles Account. For mileage accumulation, the Delta.com Account and PNR are irrelevant.