Not sure what breed of duck you have,but you will soon know which duck it is when they are through laying the eggs and begin to sit on them fulltime to incubate them.They usually lay 1-2 eggs per day. The breed I have, the mallard laid 11 eggs ( I think they usually lay 8-15). After they lay the last one she will sit on them most of the time only getting up briefly for a quick swim and to get food.Maybe you could get a bird book or get on the internet to research your particular breed to see if there are different features of the males and female so you could narrow it down.
Not sure on the eating the eggs and swelling question so I will defer.

Thanks for responding. I do know I have 4 mallards, 2 boys, and 2 girls. I have 6 Pekins. Then I have on grey one? Two black ones that are irredescent(sp) blue/green in the sun, and last but not lest I have two brown ones. No one is staying in and sitting on the eggs. We have been getting one egg a day for 7 days now. All the duck go out a swim on the pond, with out bother to care for the egg.

I can't tell duck eggs apart, but if you incubate them you'll find out the mom. LOL. Of course, they may cross breed. The eggs taste very similar to chicken eggs and even though fertile, they don't start to develop until incubated (either by the mother duck or an electric incubator). If you pick them every day, they will not start to develop. And if you refrigerate them right away they will not develop. We usually put our eggs in the fridge within a week of laying (but picked every day).

The swelling in the neck is probably just food in their crop. Ducks and chickens eat as much as they can as fast as they can and store the food in their crop, until it can be ground up by their gizzard and passed onto their stomachs. Feel the area before and after feeding.

Also, Pekins are not known for their "motherly" instincts-they don't frequently want to sit on their eggs.

Thanks Too_Many_Pets! I was thinking the eggs were Pekins because of the color.
Thank you everyone for your responses.

Well we made omlets today with the duck eggs, the kids loved them MMMMMM MMMMM good.