I can't answer with specific reference to the Roman Empire although I suspect that much of my answer is transferrable from Medieval skills to Roman skills.

Firstly I'd like to challenge the assumption that "knowledge and tools for dog training were much less than we have today". A good dog trainer needs no tools other than an understanding of the animal, there's no reason to suppose that people of medieval times understood their animals any less than modern non-technological societies do today - and they do so very well indeed. Veterinary understanding has certainly improved but this arguably has little to do with the skills required for animal training.

The types of dog used in Medieval hunts are very similar to types that we'd recognise today with the exception of modern "gun dogs", a type that has come into usage for the firearm hunting that largely replaced hawking.

Medieval hunts would use Greyhounds, a fast, short-burst game tracker that was safe in domestic settings. Alants were also used, they were a larger, stockier greyhounds for chasing larger game like bear but were unpredictable and considered too dangerous to roam free around homes. Mastiffs were used as hunt dogs and quite often as guard dogs, this large and impressive breed was used on the same large game as the Alant and also as a guard/watch dog. Mastiffs were used as war-dogs, although their usage wasn't common in Medieval battles due to their negative effect on horses, you'd be more likely to see a mastiff used by small armed patrols as a first line of defence/offence.

Running-hounds were used as game-chasers, the breeds were largely similar to the foxhounds of today.

How well trained were they? From the ubiquitous use of the different types as an integral part of hunts in Medieval England and Europe it seems highly reasonable to presume that they fulfilled their task very well and therefore that they were trained very well, particularly breeds that were required to silently find scent for the hunting party.

In short: there are a lot of reasons to think that Medieval animal training was as good as it is today - if the use of dogs spoiled the hunt then their use would have ceased rather than perpetuating for hundreds of years.

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^(Sources:)

^(Royal Forests - Hunting and Other Forest Use in Medieval England; Hooke, D; 2011)

^(Hunting Law and Ritual in Medieval English Literature; Marvin W; 2006)

^(Hunting for the Anglo-Normans; Zooarachaeological Evidence for Medieval Identity; Sykes, N; 2005)

^(Special mention, well worth looking up as a contemporary reference:)

^(Livre de Chasse (Book of Hunting); Gaston Febus, Count of Foix; c.1388)