When do we have to consider them together and not as separate regression models to be found. How is it detected ?
If you're predicting height and price of a building at the same time, you "might" be able to do better if you combine them. The variation, the gaussian error you assume in linear regression, in these two outcome variables are not independent. If it's higher than you expected, it's probably also more expensive.
You can detect that by considering the relation between the residuals. If the correlate or have some other type of relation, there might be an opportunity to leverage that.