“私下签订的”可以翻译为“privately signed” 或者 “signed in private”。这两个表达都准确地传达了“私下”进行签订这一含义。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 between the two parties. （这份合同是双方私下签订的。）