There are multiple parts to this:

1. Flights from the USA and Canada to Israel
There is secondary screening for all departures.
- El Al: document check by El Al's own staff.
- United: wanding and swabbing (at least in Newark; I don't know about Dulles and San Francisco). This is conducted by private contractors.
- Delta: walk-through metal detectors and x-ray machines. This is conducted by private contractors.
- Air Canada: swabbing of hand luggage, hands, and waist. This is conducted by CATSA.

2. Intercontinental flights to the USA
- Wide-ranging versions. These are conducted by contract security staff (I-SEC, ICTS, etc). In some cases, American Airlines staff conduct their own.
- On US carriers (usually): document check and interview before check-in, the same prior to boarding. The interviews range from brief and pro forma, to offensive and detailed.
- Some airlines have 100-percent baggage inspections; some are at random or for passengers designated as "selectees" (SSSS)
- A few airlines swab shoes and hands for explosives; others have wanding or frisking
- Secondary inspections are conducted at the gate in most cases, but some airports conduct this at the screening checkpoint (Frankfurt, Canadian airports.) At Kevlavik, there is a separate room, a separate facility, just for passengers designated for secondary screening.
- Interviews are centralized at Amsterdam/Schipol, and--often--at Frankfurt, Munich, and Zurich.

3. Flights departing the USA (not to Israel) or domestic flights within the USA
- The TSA conducts random screening at departure gates. This has been scaled back tremendously, but it is still in force. The decision to screen a particular flight is presumably based on staffing, logistics, and timing rather than any particular threat. This can range from passport and ID inspections, explosive testing, hand luggage searches, and more. Swabbing and wanding seem to have been (mercifully) ended.
- US Customs and Border Protection (CBP) conducts passport inspections and interviews on some international flights departing the USA. They are generally seeking undeclared currency.

4. Passengers with Onward Flights to the USA
Icelandair (at London/Heathrow) and some other carriers conduct additional interviews and document checks at the passenger's point of origin. This would apply to a passenger travelling--say--from London to New York via Keflavik. The interview and screening is conducted in London, not Kevlavik.
[MENTION=856805]LeoNYC10[/MENTION], that is a fascinating question.
Here's my guess: contract security agents are all over the airport: on the apron, guarding entrances to SIDA zones, guarding aircraft. I'm guessing that it is these same agents who are assigned to the Delta flight at JFK. In my experience, they're incredibly friendly. The same in Tel Aviv: cheerful, hospitable, no hassle.