You ask about your friend purchasing you a ticket from their account. If they are paying in cash, there is no impact or point of the purchasing party having logged in into their account. Once the ticket is purchased, they'll get a record locator and other details of the ticket. If they add your skymiles number, it will show up under your account when you log in. If they didn't add your skymiles number, you won't see any details of the ticket in your account. As others have pointed out, you can search for the ticket but you'll need an additional piece of information to find it (record locator, ticket number, or credit card number used to purchase the ticket).

If on the other hand, your friend is purchasing your ticket using DL skymiles (i.e., purchasing an award ticket), they will have to log in into their account in order to complete the purchase. As noted above, they'll have to add your skymiles number for the ticket to show up in your account as noted above. The other thing to note is that you won't receive any skymiles from this trip since award tickets do not earn miles. You will, however, have the associated benefits of your skymiles membership so it would still make sense to add your skymiles number to the record.