[throwaway568]

You can get into the iPhone or Android and copy the SQLite database to your computer.

The images can be tricky - some form of reference to the image in your phone, hard to put together; otherwise the messages in all the chats are in the database, navigable by a SQLite client.

Speaking from personal experience porting messages from iPhone to Android.

[RobertSponge]

The backup is encrypted. The private key can be extracted if your android device is rooted.

[2rsf]

Isn't it encrypted ?