Let's not think "その can mean you(r)". Actually you can say あなたのその目で確かめよう, which is perfectly natural, and a word like あなたの is usually omitted in Japanese anyway.
This その has some distinct effect. その is typically used when a speaker is actually seeing something near the listener. For example, an angry person may say (お前の)その耳は飾りか? ("Are those ears (of yours) just for show?"). So this use of その makes the message sound more dramatic because it sounds as if someone is seeing YOU and conveying this message directly to YOU (imagine that "I want YOU" poster). To take another example, a warning message for ore-ore fraud may be like その電話は本物ですか, which sounds more dramatic than あなたの電話は本物ですか.
