“私下签订的”可以翻译为“privately signed” 或 “signed in private” 。
“privately signed” 强调签订行为是在私下、不公开的环境中进行的。
“signed in private” 同样表达了在私人、非公开场合进行签订的意思，二者语义相近，可根据具体语境选择使用。
例如：
They reached a privately signed agreement.（他们私下签订了一份协议。）
The contract was signed in private by the two parties.（这份合同是由双方私下签订的。）