Training a model where each response in the observation data has a different known varience
I may be understanding the question now.  Still using the coin example, as I said above, the number of trials for a given coin only affects the confidence for the estimated probability/bias for that one coin.  So it seems like you are asking how to incorporate that "confidence" into the response variable, if at all.  In other words, you are asking if your model should reflect the uncertainty concerning the true value of $p$ for each coin, given the number of coin flips you performed.

I don't think assigning different weights to observations is appropriate in this situation because, again, the number of flips for one coin does not have anything to do with other coins.

I am not sure if this will satisfy your needs, but there is something called interval regression that is used to model a response/dependent variable that is defined as an interval between lower and upper bounds.  It is a type of regression for censored data (a problem where the true value of response is not known) and is typically used for modeling such variables as income ranges or survival times.  In your case, you could compute a confidence interval for the true value of $p$ for each coin, using the $p$ calculated from your trials and the number of trials specific to each coin.  Then you would use this regression with two response variables: lower limit and upper limit of the confidence interval.

Based on my quick search, I am not finding a lot of Python support for this type of model, except:

Maximum Margin Interval Trees - Decision trees for interval regression

Drouin, A., Hocking, T.D. &amp; Laviolette, F. (2017). Maximum Margin Interval Trees. Proceedings of the 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.

https://arxiv.org/abs/1710.04234

https://aldro61.github.io/mmit/