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Looks reasonable, but maybe two things which have to be considered:

1. In some use cases an explicit JSON null is considered a value, so maybe the annotation should have an allowNull (or similar) annotation element with default value false to support that use case.
2. Instead of tracking which field names have been deserialized you should probably track the Field (needs some refactoring). Otherwise this won't work for @SerializedName with alternate, e.g.:

   class Test {
       @SerializedName(value = "a", alternate = "b")
       String s;
   }

   For the exception listing missing required fields it might be reasonable to report the 'main' name, i.e. either the field name or @SerializedName#value, otherwise the exception message might be confusing.

Note that I am not a member of this project, so these are only suggestions.

@Required sounds like a twin of JSR-303 @NotNull and does not seem to make it extensible. I would not create a doubling annotation, but rather would think of implementing a generic method to apply any kind of post-processing at least at DTO class level (probably by extending ReflectiveTypeAdapterFactory that makes a lot of things hard to accompilsh being not extensible). Validation is only a particular case of such a generic thing. This all seems to be pretty achievable using post-processing mechanisms like RuntimeTypeAdapterFactory from the Gson Extras demo:

• it's possible to do anything with a deserialized object, including validation (entire JSON document must be consumed though);
• it's possible to incorporate any strategy (including JSR-303 validation as a Jackson-like module that can be built on top of type adapter factories) right into type adapter factories, so that any ReflectiveTypeAdapterFactory.Adapter-derived type adapter could invoke the strategy and quick-validate the just-deserialized object (subobjects revalidated every time a super object is completed to be serialized, so it may be time expensive -- but it sounds like it can be optimized greatly).

Any other thoughts on making Gson deserialization-only, but extensible on user demand only, tool?

UPD1 (just not to spawn another comment): JSR-303 annotations are runtime-visible and have nothing to do to static code analysis and Android/JetBrains at all.

but rather would think of implementing a generic method to apply any kind of post-processing

The problem with the post-processing approach is that it, in case of a field of a primitive type, it won't let you tell the difference between the lack of that field in the json and its presence with its default value (0/false). Yes, boxed types can kinda awkwardly solve this by allowing those fields to be null. But you won't want to use boxed types in all your objects just so you could validate them in a post-processing step. You really want to do this in a place where you have access to the JSON.

JSR-303 @NotNull

This is the first time I've seen this after more than 10 years writing Java. I do know about @NotNull/@Nullable commonly used in many places these days, but these usually either come from a JetBrains library for Java SE or an AndroidX library for Android. And — I just looked — their retention is "class", so not visible at runtime.