In a Watson's conversation to keep the context, do I need to send only the conversation_id every request? or the entire context object?
You need to send back the conversation_id and the system part of the context. The conversation ID uniquely identifies the specific conversation. The metadata in the system info has details about how long the dialog has been going on (how many turns) and in which dialog node you are. See here in the IBM Watson Conversation service docs for more information about maintaining state.
The context object also includes the user-specified data. That part can be wiped out or newly set on each turn.
What you can do is to first copy over the entire context object, then manipulate the user-oriented parts.