Might be worth taking a close look at the flexible fuel lines in the engine bay.
Up to the late '70's Fiat used a plastic hose usually blue or green? in color. This would harden over time but usually didn't leak unless disturbed, it had the maker's name Cavis at intervals along it's length along with the size e.g. 6 x 8 or 8 x 10 (iirc).
Fiat then changed to a black braid covered rubber type hose. This could appear sound on the surface but the rubber under the braid could become porous. I remember seeing these hoses 'sweating' with fuel, especially with the engine running and the fuel pump working. They should never feel 'wet'.
I've heard that modern fuel containing ethanol is very hard on these old type hoses. Best to change them for modern ethanol-proof hosing.
On the carburettor engines, there is a small drain pipe under the inlet manifold, directly below where the carb. is mounted. It's a small pipe maybe 3mm in diameter with the end almost closed down, so maybe a 1mm hole. It's attached to the manifold with a 8mm nut which is part of the pipe. When a hot engine is switched off, heat can soak into the carb. and heat the fuel in the float chamber causing it to expand and rise in level. This rise in fuel level can result in some fuel spilling from the carb into the inlet manifold where it 'should' drain out through this little drain pipe. This fuel spillage can easily happen if the float valve needle and seat is worn (to be expected on an older car). Some versions of carb. had a vent flap on top of the float chamber to allow heat to escape and overheated fuel to evaporate. This flap opened at idle (or with the engine turned off) but closed if the throttle was opened. Idk if this type of carb was ever fitted for the U.S. market, I don't think it was a very safe idea!
So, to summarize, what you're experiencing may be quite 'normal' as someone else has said above but I'd still check out and if necessary, replace the flexible sections of fuel line within the engine bay. There's also 2 sections of flexible fuel line down at the fuel tank and likely (I'm not very familiar with the Spider but am with the Sport/Coupe) a large rubber pipe/elbow linking the fuel filler to the tank, now this gave a lot of trouble..
Carrying a fire extinguisher is also a very good idea when driving older cars, particularly if you don't know the condition of fuel hoses etc.
Al.