Non-parametric regression on set of time series: One model for each or one for all series?

Let's say I have a set of 1D time series which values have been samples in equip-distant time steps with timestamps $1,2,3,...$, they have all the same lengths and are somewhat similar in shape. I want to apply non-parametric regression (e.g. with Gaussian Processes or Kernel Regression) on the time series in order to infer values for timestamps that are between sample timestamps (e.g. $5.3$).

The obvious way of doing this would be to simply build a regression model for each time series. Overfitting, if a single regression model is build for each time series, can be avoided by using established techniques in using the right smoothing parameter in the respektive kernels.

Is there some game-changing merit in taking the entire time series data and applying the regression model to the entire time series dataset? So far I only see the disadvantage of over-smoothing (aka underfitting).