How can multiple sites use a common default theme with site specific customizations? We are using Aegir, we would like to use one core theme for all of our websites which handles common theming. But for specific theming for each site, we would like to handle inside of their custom themes.

Basically this will enable us that we can change common theming in just one theme used by all, instead of making the same changes in every theme of respective site.

One challenge by creating the custom theme in one of websites it will overwrite code from common core theme, but both themes should be present in the same time.
Is there any kind of solution for something like this?
Consider starting from a Base theme, such as Omega, Zen, Bootstrap, etc. Or build your own (custom) theme that you can use as a Base theme (that would be youru "core theme".

And then create Sub-themes (that would be what you named "their custom themes").

Note: the question about "Right way to start with a Bootstrap subtheme?" contains a lot of details to get started with such sub-theme ...