Nobody should ever share a private key regardless of the software or specific use. That’s why the key is called private.
In order to open a Word file, you need Microsoft Word or compatible software installed.
In order to open a PDF file, you need Reader or compatible software installed.
In order to open a TurboTax file, you need TurboTax or a compatible software installed.
In order to open an AxCrypt-encrypted file, you need AxCrypt or compatible software installed.
So, in order to share an AxCrypt-encrypted file the other user must use AxCrypt (or compatible, but none is available as far as I know).
What you also do as a Premium user is to use the key sharing functionality (this is not the private key, it’s a different key – often called a session key) of AxCrypt by entering the other users email address to share with as recipient. Once that is done, you can send the file to the recipient who also needs to install AxCrypt (but does not need to be a Premium user). All this is explained and shown in videos and other material available at https://www.axcrypt.net/ .