radioactive decay is more about the isotope involved than anything else ... Cobalt 60 (Co60) has a much longer decay half-life than some other materials. Half-life being the time it takes for it energy level to reduce by one-half. For Co60 that is measure in years. For other things its measured in hours, days, months etc.
There are other long-lived isotopes that are resultant from a nuclear detonation .... and furthr ...if you are talking "dirty bomb" there is no "standard" makeup ...and whomever it was said it first ..is probably ritght. Dirty bombs are more about contamination than radiation. The bomb is dispersing hi level contaminants into the air to spread around for us to get all over ourselves and inhale or ingest so they can deposit inside us. Low level comtamination alpha and beta can be shielded by our sking ...but if you ingest it (breath or swallow) it irradiates our organs from the inside causing damage and cancers etc. But depending on the bomb makeup it can contain activated material isotopes with an extremely long half-life ....and one can't tell without some sophisticated instruments to anaylze them. So, what to do ? Defend all in the same basic manner...shield yourself. create decon and clean zones so that once you do venture out you make sure to decontaminate ...clothes and people before re-entry into the clean zones. This can be done by removing the outer layer of clothing or all the way to full-fledged showers. That's why you see Hazmat guys in suits ... the suits gets "crapped up" and are discarded or left "outside" as part of re-entry into the "clean zone" of your shelter. Either way for that you'll need at least the basic radiation monitoring instrumentation to make at least a token attempt to see that you are "clean". These instruments are available on the surplus market and represent technology that's 20 years old or more ...but they still do a good job and are reasonably priced.
If you are going to be in or want to venture back to a bomb area or one that got dusted with fallout you will need this technology if you are at all concerned about radiation hazards ...since you can't see it or smell it and once its affects are clearly visible its already too late for rudimentary corrective actions.
Some fun awareness training has been done for teaching about contamination and the spread of it, by dusting a particular area with a powder that will flouresce (spelling ?) under a black light....even once people are shown the right way to act they forget and when you turn on the black light you can see every place they touch anything after first touching the "contaminated" areas. It sounds scarier than it is ... staying "clean" is a lot easier than "getting clean" and being able to detect contamination isn't hard if you have the basic tools.