I have done both.
My son's birthday is in April. Every so often, it is near Easter, so one year, I arranged for a neighborhood Easter egg hunt/pizza party. It spanned 3 yards and was super fun! I bought 12 eggs for each kid at the Dollar store and Easter baskets, there were 12 kids. I had also bought 12 larger eggs with a bigger toy, like a matchbox car or ring. Each kid got one. (like a party favor!) and then had some family members help me hide them.
The I stood in the street, the kids on the edge. I would call out 6 colors. Then I said find 1 BIG one. After that I said, okay, there are 6 more eggs for each of you...GO! This helped spread the hunt out a bit and was funny because if they found the wrong color, they couldn't pick it up. hehehe. As the kids were eating pizza, I brought out some cookies and cupcakes and said, oh by the way, today is C's birthday! Let's all sing Happy Birthday!
For my daughter, I had a Halloween party, same set up.
The key is to not call it a birthday party, just a get together, or pizza party. If you say birthday, you will get presents.