If the FBI subpoenas a person's credit record, the target of the investigation won't see that on his credit report. Similarly, if a person's emails are being monitored, the target won't see any indication of that.

If the target of an investigation is under physical surveillance, it is possible that the target will detect this. Physical surveillance is expensive and relatively rare.

A person's residence abroad complicates investigations. However, the United States has Mutual Legal Assistance Treaties (MLATs) with numerous countries, and the FBI has legal attaché offices at some U.S. embassies:

https://www.fbi.gov/contact-us/legal-attache-offices

If the target of an investigation is using an American phone service, email provider, or messaging app, the FBI could directly subpoena the service provider. Assistance could also be sought from foreign governments.

In addition, if a target's mobile number is known, technical means exist to monitor that number's calls, texts, and location data exist owing to weaknesses in the SS7 protocol used in cellular telephony:

https://en.wikipedia.org/wiki/Signalling_System_No._7#Protocol_security_vulnerab...

In short, if a person is under criminal investigation by the FBI, that person probably won't know about it until he or she, or an acquaintance, is contacted by the FBI.