As already said, keep the saw in the truck...along with gas, spare chains, scrench, fuel, gloves, and oil. when using the trailer, I keep an 8' 2x4. That thing has been used many a time for various reasons.
On your example of the logs on the shoulder of the road, I woulda grabbed 'em being state property and that close to the road. I would not have cut them there....throw and go baby....throw and go.
As far as the orange safety gear to the above poster, I'm not laughing. I have a vest, and 2 reflective highway cones. I am not afraid to stop on the busiest of roads and cut wood.
Other gear I keep in the truck: AT LEAST 2 saws, files, come along, tow strap, chain, 4' pry bar, wedges, gloves, gloves for as many passengers as your truck holds

, ear protection, business cards to stick in peoples doors where you see downed wood in their yards and knock but get no answer...offering free removal and piling of brush on their property, proof of commercial liability (may not apply in your case-I have to have it anyway for my business), always have a case of water in there, 2x10 to use as a rolling ramp for the big rounds, and blank liability releases if your not carrying liability insurance.
Now for a couple tips I've learned:
1. Dont go to far. 20 miles is about my max unless I have seen pics of the conditions, tree(s), access, etc. and I know it's at least a cord. Gas in the truck while pulling a trailer isnt cheap, plus your time is worth something....no matter what you mother in law thinks.
2. Dont rely on the homeowners to id the tree for you. If you dont know trees, get a book for your reigon. I reccomend the national audobon society ones.
3. Only cut what you can haul that trip. If I drive by, see you cutting, watch you leave, and you leave a bunch of pre-cut wood laying there, who do you think is gonna have the wood you spent time and money cutting? Hint, it wont be you.
4. Wheelbarrows are your friend for small to medium stuff. Use your ramp and dump....then repeat.
5. If you forget or dont have a ramp, back the truck into the closest ditch.....itll drop the tailgate height dramatically saving your back. If need be, drag the damn log with the truck to the ditch to save even more time.
6. After storms in your area, post CL ads offering free storm clean up. Be ready to work. The phone will ring.
7. Once you get 2+ years ahead on your wood, then you can be picky as far as species go. Until then, if its not rotten, grab it. It'll burn once dry.
8. DO NOT get in over your head. You will see what I mean once you post your first storm clean up ad on CL. You will be tired of cutting wood and answering phone calls from clueless people who want their standing trees cut that are hanging over their trailers, homes, sheds, tents, powerlines, kids, dogs, goats, pigs, chicken coop etc.
9. ALWAYS make sure the person you ask for permission actually owns the property. Tenants dont count--they may lie to ya and you wont know it.
10. Call local tree services and city dumps in your area. They normally are happy to get rid of wood. The worst they can do is say no.