watching each other / keeping an eye on each other / closely observing each other
there is a system where they have to keep an eye on each other
there is a system where they have to keep watching each other
「お互いを監視する」＝ watching each other / keeping an eye on each other / closely observing each other
「お互いを監視し合わなければならない制度がある」＝ there is a system where they have to keep an eye on each other / there is a system where they have to keep watching each other
In some countries, people are required to keep an eye on each other.
こんにちは！
ご質問ありがとうございます。
先のアンカーさんが回答されておりますので、ここでは他の例を挙げてみます。
In some countries, people are required to keep an eye on each other.
とすると、「とある国ではお互いを[監視](https://eikaiwa.dmm.com/uknow/questions/79690/)し合わなければならない制度がある」となります。
役に立ちそうな単語とフレーズ
surveillance 監視、監督、見張り
参考になれば幸いです。
イギリス 
日本 
