Theoretical background for model arhitecture choosing
The theoretical estimation of the error depends on the data, and the fact that the labels can be predicted by the variable X. There's no way to a priori know which will be the best option regarding the two models, so just try.

However, I would bet that the second option will work better if y2 is correlated in some way to y1, and I would bet for the first option if they are independent.

However, when you train your model, I highly recommend sampling the data in a more balanced way, so that the model not learns to predict (0, 0) always regardless of the values of X (as a model that does these dummy predictions would have high accuracy). Another way to work around the imbalanced classes is to use AUC in ROC as a metric to choose among different models.