How old is she? Do you know when she last laid?
When a hen has a swollen abdomen and a consistently dirty rear, it's often a sign of a reproductive problem. If it feels water balloon like, then it could be fluid, called ascites. That is caused by the liver leaking fluid into the abdomen. It accompanies various issues like cancers, infections, organ failure. Most often they will stop laying eggs. The pressure from whatever is going on makes it harder for them to push droppings out and results in dirty rears. If needed you can trim back the feathers under the vent to help keep droppings from building up. Those droppings put them at risk of flystrike, from flies laying eggs there.
A firmer swollen belly could be from internal laying or infection like salpingitis. It's often hard to know until necropsy since symptoms can be very similar. Treatment options are very limited, and often not successful. If vet care is an option then some imaging may be able to tell you for sure what's going on.
