Innocent times spent sharing documents and executable files without a care in the world.

以上句子可以按照第1种方式处理后为完整句子，也可以按照第2种方式去理解为一个名词短语

(1)加一个助动词"was"构成一个完整英语句子如下所示

Innocent times was spent (in)sharing documents and executable files without a care in the world

(2)可以看做一个定语从句省略“关系代词that+助动词was”结构名词短语

Innocent times spent (in)sharing documents and executable files without a care in the world.(过去分词短语作后置定语，修饰前面的名词“innocent times”

=Innocent times that was spent (in) sharing document and executable files without a care in the world(定语从句作后置定语，修饰前面的名词“innocent times”

翻译：在那样的纯真年代，我们共享文档和可执行文件，根本没有半点顾虑。

解答如下

以上不是一个完整的句子，而是一个过去分词作后置定语修饰前面的名词的一个“名词短语innocent times”

请问真的不是一个完整的句子吗？可是，它是用句号结尾的

继续跟踪回答如下：

以上句子结构为下面句子结构被动式

主动语态结构：主语(sb)+spend+时间/金钱+(介词in)+doing sth

I spent two hours in finishing my homework.

被动语态结构：时间/金钱+be spent(介词in)+doing sth (by sb)

Two hours were spent in finishing my homwork.(完整的句子）

Two hours spent in finishing my homwork.（带有定语的名词短语）

=Two hours that were spent in finishing my homwork.（带有定语从句的名词短语）