The Bot class has a log and debugLevel variables that control logging. You can set these on your Bot instance in code.
There are methods to import chat log, response list, and AIML on the TextEntry class. But there are no methods for exporting a conversation or responses. You could implement such as method in your own interface, either capturing the conversation, or querying to bot's memory for #conversation objects or #sentence objects that have a #response.
I would recommend using a response list file over a chat log for training, as it gives you much better control over what the bot learns. Chat logs are more for 'free' learning, than training.
The difference between importing AIML as a file script and as a log is that loading as a file script creates a script/state object in the bot's language thought. States are executed precisely and will obey AIML semantics. Importing as a log will not create a script or state, but import the AIML categories as responses in the bot's knowledgebase. This lets the AIML to be used with any other responses the bot is train with, and allows for heuristic matching of the patterns. This means that the pattern "Do you like hockey" might also match the question "Do you love hockey" (the bot will find the best match).
For a set of example response lists and scripts see,