Weird distribution of neural network outputs
It's not that weird. You probably have an activation function on output that only reach 0 or 1 at infinity. To reach that 0 or 1 you would need to have your last hidden layer to output +/- infinity. Which is difficult given the weight structure. The best it can do will be to output extremely large positive or negative values.
It may be a problem because during the calibration process (which is an optimisation problem) you will spend significant ressources and time in trying to reach +/- infinity. Notably this is one of the causes of the famous exploding gradient problem.
How to deal with that ? In Efficient Backpropagation (LeCun and others, 1998), the author propose to use an activation function that is able to reach desired outputs. It states that a good activation function needs to have the following properties :

$f(\pm1)=\pm1$

The second derivative is a maximum at $x=1$

The effective gain is close to 1


Your problem is that you don't have something similar to the first condition. To respect those conditions LeCun proposes to use the activation function: $1.7159*tanh(\frac{2}{3}x)$. you might not need to use it in all your layers. Now your problem would probably be to find a framework that allows for custom activation functions.