The best I can say, is that there are various studies regarding the anti-inflammatory and analgesic properties of all the allium family, including the scallion.
analgesic and anti-inflammatory effects of the Allium Ascalonicum (scallion)
In conclusion this study has shown that the acqueous and methanol
extracts of Allium ascalonicum have mild analgesic activity and strong
Antioxidant, anti-inflammatory, and antimicrobial properties of garlic and onions (abstract only)
Findings – Both garlic and onions exert their effects on human health
via multiple different functions, including antioxidant,
anti-inflammatory, and antibacterial properties. The organosulphur
compounds in these spices scavenge oxidizing agents, inhibit the
oxidation of fatty acids, thereby preventing the formation of
pro-inflammatory messengers, and inhibit bacterial growth, via
interaction with sulphur-containing enzymes.
Smelling, so inhaling the fumes of onion and garlic is essentially inhaling the volatilized chemical componds, expecially if the scallion or onion is minced and warmed.
That's not very different from the way commercial products like Vicks VapoRub works, as the "Vapor" part is essentially inhaling menthol and camphor fumes, and both have analgesic and anti-inflammatory effects.
Those analgesic and anti-inflammatory properties are going to help with the symptoms of the cold (as Avrohom Yitzchok has said, right now you can't really cure the cause of the cold).
Adding to that, the anti-microbial and anti-bacterial properties may help curing eventual bacterial complications caused by the cold.