How is MNIST only providing the training and the test sets? What about the validation?
In order to perform cross-study comparison of model performance it makes a lot of sense to have a single test set for benchmarking. This way, different investigators can compare their respective models in an "apples to apples" manner. The test set is tested only once, as already mentioned, and is in a practical sense the final arbiter of model performance. As an aside, the MNIST not only provides an important source of data for testing individual hypotheses, but it also serves to provide "standards". After all, NIST stands for National Institute of Standards and Technology.
The reason why a validation set is not parsed out (like the way the test set is).
Different investigators may want to perform different types of validations (hold out, n-fold cross-validation, leave-one-out cross-validation). Thus, MNIST does not limit investigators from doing this by separating out a validation set. MNIST leaves it up to the investigator to parse their "training set" into yes, a training set and validation set they way they prefer. Unlike the test set, there is really no need to standardize the validation. In contrast, test set does need to serve as a standard for benchmarking models.
NOTE: To be clear, this does not mean that you should not perform a validation. You absolutely must. The only thing is that you must "carve out" the validation set in a manner of your choosing.