We don't know the numbers sure, but we have some descriptions.
The Dominion fielded two armies consisting of 'all available forces.'
The Army in Hammerfell was crippled by the legions before they left Hammerfell and then driven across the desert taking heavy losses by the Alik'r.
All this happened prior to the battle of Red Ring.
We're given no real indication of where the Alik'r stood troops wise, but given they eventually won on their own without further imperial assistance it's reasonable to assume "Better than the Dominion."
Then there's the battle of Red Ring, where the larger Cyrodiil invasion army was entrapped and entirely destroyed.
The cost to the legions is said to be three legions of at least 8 wiped out entirely, and the remainder with heavy losses leaving the best at half strength.
In the aftermath of Red Ring, the Empire sues for peace with the white gold concordat.
Hammerfell rejects the concordat and after five years of fighting without imperial support pushes the Dominion out of Hammerfell.
That doesn't indicate that the Dominion had the steam to actually put up a fight.
It's also interesting but obviously non-specific that the Imperial losses at Red Ring and the Aldmeri losses in Hammerfell at the hands of the Alik'r are both described as 'Heavy' implying some level of equivalency in losses as part of their overall strength.