The network is already doing its best at minimising the distance. If your final goal is point estimates that minimise the distance, predicting the error is probably not a good way to go about improving performance.

However, if you care about the uncertainty / having a distribution over where the ground truth might be, then there are definitely various techniques that allow this.

For example, if you expect the errors to change depending on some conditioning variable, you could have the neural network output the locations (mean), and the standard deviations (uncertainty) of the positions, given the conditioning variables. In practice you would output log stds and exponentiate to ensure positivity. Then you could use a Gaussian likelihood function approximation, replacing the L2 norm with the negative log likelihood under the Gaussian assumption.