How about if you define your loss function to be
if continuous_label1 & continuous_label2:
("MSE loss for continuous_label1" + "MSE loss for continuous_label2") / 2
else if continuous_label1 & not continuous_label2:
"MSE loss for continuous_label1"
else if not continuous_label1 & continuous_label2:
MSE loss for continuous_label2"
skip training example
"if continuous_label1" is to be read "if continuous_label1 exists"
EDIT: instead of dividing by 2, it is better to use the weighted average