If you want your eggs to be speckled, as many songbirds' eggs are, find a spot where you can get a little messy with acrylic paint. Work outdoors or over a sink or lined wastebasket-- anywhere spatters of paint won't be an issue or can be quickly and easily cleaned. (Also, you will not want to do this step in your nicest clothes.)
Dip a toothbrush into a little acrylic paint. (You can squirt a few drops out onto a bought or makeshift palette or work directly from the bottle or tube.) We used paint in a few shades of brown and beige, but feel free to experiment.
Practice spattering the paint onto a scrap of paper until you get the hang of the technique. Draw a finger or your thumb back over the paint-laden brush. Notice how the paint spatters (mostly) in the opposite direction.
When you're comfortable with spattering, exchange the scrap paper for your cured and cooled eggs.
You may find it easiest to spatter your eggs in stages, allowing one side to dry before spattering the other. Continue spattering until your egg is speckled to your own approval. You can even spatter multiple colors onto the same egg, for greater depth and variety. For less opaque speckles, try diluting the paint with a very little water before spattering it. You can also "splotch" your eggs by randomly dabbing paint onto them with a fingertip or brush.
Note: The point of spattering is to create a random speckling, but if you are dissatisfied with your results, you can always remove the paint and start over. If the paint is still wet, wipe it away with a damp cloth or paper towel. Rubbing alcohol, soap and water, or a wet wipe usually do the trick. If the paint has dried and is being stubborn, you can sand it away with a fine grit wet/dry sandpaper (400 grit, for instance).