I've gotten speckled eggs from various breeds of hens. The most recent speckled-layer I had was a Production Red (a lot like a Rhode Island Red.)
I would say any breed that lays brown eggs can have individuals that lay brown-with-speckles eggs.
You can spend a few days sitting in the chicken pen watching them go in and out of nestboxes and lay eggs, or you can set up some sort of a camera to film who goes in the nestboxes, or you can go look every few minutes all morning to see which chicken and which eggs are in what nestbox.
Or if you have a cage that can hold just one chicken, plus food/water/nest, you can figure out who is laying them. Just shut one chicken in the cage in the morning. If she lays an egg, you know what hers look like. If she does not lay an egg--see whether a speckled egg was laid by any other chickens that day (which means she did not do it.) Once the hen in the cage lays an egg, or once any chicken lays a speckled egg, you can let that chicken out of the cage. Try another one the next day, until you have it figured out.
You can check their vents to see who is laying, because the non-layers are clearly not responsible for the speckled eggs. Just look at each chicken's vent, the hole that eggs and poop come out of. A rooster or a not-laying hen has a small, puckered vent. A hen that is laying has a vent that is larger, looks somewhat moist, and looks like it could stretch to let an egg out. The difference is pretty obvious after you've seen both kinds.