"She wants to drink her water."
Translation:Cô ấy muốn uống nước của mình.
Of course, that is her water so "Cô ấy uống nước" is good enough. Unless that water belongs to someone, Vietnamese doesn't need to say it out.
Regarding "Mình": "Mình" means myself/yourself/herself depends on the subject I/You/He/She. "Mình" is very subtle and cute meaning that takes pages to explain. For wives and husbands, they may call each other (old fashion) "Mình" to show their united/unified body also.
In this example, if it says: She drinks our water then it should be translated to "Cô ấy uống nước của tụi mình/chúng mình."
Vietnamese is very concise and precise language. In other words, each language has its own structure, if you want to speak like the natives, please don't try to force other language structure into your own.