Being the SkyWarn Director for Macomb County here in Michigan and former Director in Davidson County, Tennessee, I can give you this information about SkyWarn:
ALL NWS offices have skywarn training. These classes are offered to a variety of people. Mostly, these are the general public and amateur radio operators. The "Hams" mostly are affiliated with their local "A.R.P.S.C." organization (Amateur Radio Public Service Corps.). These groups are sponsored by the local county (and sometimes city) governments and provide supplemental emergency communications at various times for certain emergencies or events. SkyWarn in these counties (as far as hams go) are under the "supervision" of A.R.P.S.C. When the Hazardous Weather Outlook (or HWO as it is referred to) states - "Dispatchers, notify your emergency managers immediately" - one of the people that are notified are either the EC for the A.R.P.S.C. group and/or the SkyWarn Director.
Now, when the HWO says "spotter activation may be needed", it is this group of spotters that is its intended target. When the Storm Prediction Center (who issues WATCHES) issues a WATCH for a particular area, most A.R.P.S.C. groups in that area put their skywarn program into "stand-by mode". This simply means that the spotters in that area have been notified of possible severe weather in their area sometime in the period stated in the watch. Hams, of course, use radios for their activities and almost all NWS offices have an Amateur Radio Station set up in their office. In most cases, then, there is a network of linked repeaters that are connected and given a certain name or acronym - here in Michigan, it is called M.I.C.O.N. - Michigan Inter-county COmmunications Network and in Tennessee it was called M.T.E.A.R.S. - Middle Tennessee Emergency Amateur Radio System - and these systems or networks are also put into stand-by. When the possibility of severe weather is very high, the network is "activated" by the NWS by several means. Usually though, people affiliated with this network are either text notified or paged that the NWS office is requesting spotter activation.
These radio "nets", as they are called, are considered by the FCC as "emergency communications" which is why most county skywarn programs fall under the A.R.P.S.C. organizations supervision. Let me be clear about this though - SkyWarn is a program of the National Weather Service - NOT the local group. Hams assist because we are much better prepared and capable than the ordinary citizen to give "real-time" weather observations because we are volunteers, are NOT compensated nor can we be (per FCC rules) and most of the equipment that the NWS office has for amateur radio is either donated or paid for by grants and this money does not come out of the NWS office's budget - amateur radio is NOT a cheap hobby!
All this said, I can also tell you that Nashville, TN police, fire, their dispatchers and DPS workers are all required to have skywarn training because they are in the public service sector and on most days are out in the weather as it is.
So, if you want to participate in the skywarn program on an "advanced" level, per se', it would be wise to become a ham and find your local A.R.P.S.C. group or whoever is running the skywarn program in your area. The NWS will be able to give you that info and who to contact.
If you need anymore information, you can contact me direct!
Hope this helps!
Dennis - N8BMB