The storage of livestock manure is responsible for ammonia emissions into the atmosphere. Different natural covers could be used during animal manure storage, but the mitigation effect is influenced by the manure characteristics due to the housing or treatment systems. Starting from cattle and buffalo manure, the objectives of this study were (i) to assess the effect of anaerobic digestion (AD) and solid–liquid separation (SLS) on ammonia emissions during storage as well as natural crust development and (ii) to investigate the reduction in ammonia emissions by using a layer of straw to cover the stored animal manure. Storage conditions were simulated in a small-scale application in a climate-controlled room. Results showed that the higher organic matter content of cow raw slurry facilitated the surface crust formation starting from the first days of storage. AD with SLS increased ammonia emissions (48.5%) due to the increase of the ammoniacal nitrogen content. On the other hand, animal manure covered with a layer of straw showed a 7.3% reduction of ammonia emissions. This study suggests that treatments and covering strategies must be calibrated to different manure types to enhance the mitigation effect.