Wikipedia claims it is an old Russian proverb, and that Reagan was coached in its use by one of his speechwriters. Lenin apparently liked it as well, but not enough to verify what Stalin was doing. Or something...
Anyhow, Reagan and Lenin probably both ate carrots too.
The more serious issue is that the allied countries may host populations that are not friendly. Specifically, the Hamburg cell (Mohamed Atta's group) were operating out of, duh, Hamburg, Germany. Presumably, if you wanted to have a chance to intercept guys like this, you'd have to listen in on private German conversations. Also presumably, German authorities might not see plotting to do nefarious things in other countries as a problem they needed to be concerned about - see the Munich Olympics aftermath:
An article in 2012 in a front-page story of the German news magazine Der Spiegel reported that much of the information pertaining to the mishandling of the massacre was covered up by the German authorities for the past decades. For twenty years, Germany refused to release any information about the attack and did not accept responsibility for the results. The magazine reported that the government had been hiding 3,808 files, which contained tens of thousands of documents. Der Spiegel said it obtained secret reports by authorities, embassy cables, and minutes of cabinet meetings that demonstrate the lack of professionalism of the German officials in handling the massacre. The newspaper also wrote that the German authorities were told that Palestinians were planning an "incident" at the Olympics three weeks before the massacre, but failed to take the necessary security measures, and these facts are missing from the official documentation of the German government.
Relying on Germany to tell you that you are about to be attacked by people living in Germany seems to be a policy of questionable intelligence. Maybe every other country is better, but I doubt it.
If it makes you feel better, the US had a hard time tracking/suppressing IRA support back in the day. I would expect there were UK operatives in the US at the time, trying to do the work the US was unwilling/unable to do. So, yeah, "allies" will spy on each other to some degree. Being allies doesn't mean everyone that lives in your ally's country is friendly.