To answer this question you have to really first define who the average person in the Medieval period was, and who the average person in the Roman Empire was. It at first seems easy; the average person in the medieval period was a peasant, and the average person in the Roman Empire was living in a beautiful city with modern plumbing, access to water, and very likely assisted living allowances.

But, quite frankly, that second one is a fantasy. Most of the urban population of the Empire were not living in the five or six largest cities where those amazing public amenities were, and instead were in a smaller class of cities such as Londinium. These cities certainly still offered some comforts, but were by and large not that different from the cities of the Medieval period, serving as administrative and trading hubs for the far greater share of the population living rurally. Indeed, even at it's height, the urbanization rate of the Empire likely did not exceed 10% of the total population (up to seven million of a total population between 50 and 70 million), leaving the other 90% of the population living on the farms and large estates. Now, before I go on, I should say that this might lead someone to immediately think that the majority of that 90% is slaves and thus they must have had a terrible life. The answer is that this isn't really true for the most part. While slavery was very common in Italian estates, where it may have made up nearly half of rural labour, this also represented the bulk of the slave population in the Empire. The majority of rural population in Rome was made up of either paid labourers or tenant farmers who paid a portion of their crop to the landholder. I cannot remember the man's name, so hopefully someone else can remind me, but he was a fellow who left a grave marker in Roman Africa telling how he went from being a paid labourer, all the way up to managing a team of labourers, all the way up to eventually buying his own farms. However, this was likely an outlier.

So, if we assume that the majority of people continued to live rurally, and in relatively similar conditions between the Empire and the Medieval period, what are the key differences that might help us answer this question? Well, I'd say the big two to consider are taxation and security. The late Empire in particular ramped up taxation to pay for the increasing needs of the state to fight off the Persians and Germans, something that Peter Heather suggests increased to unsustainable levels as Gaul, Spain, and then Africa fell. This taxation would have meant ever decreasing surpluses for local landlords and farmers. In this sense, the fall of the Empire would have meant more local production and surplus could be retained by those farmers for feeding themselves. On the other hand, that taxation also paid for security, and there's no doubt that the lack of security in the early Medieval period had an impact on people, with raids from migrating tribes being a problem all the way up into the 9th and 10th centuries with the Vikings. However, it's doubtful much actually changed for the average Joe working the fields under Germanic leadership, as most Germans were absentee land lords in that early period. The main question is whether they would have taken nearly as much taxation as their Roman predecessors, and I'd say that's unlikely.

All this said, there is also one other thing to consider, and that is total population. Population in Europe fell drastically between 300AD and 600/700AD. When the population is high, it means a large amount of relatively marginal land is forced into service as agricultural land. With a fall in population, you have two really important changes. One is that the land being put to use agriculturally is usually on average more productive, since less crappy land needs to be tilled, and second, you actually have more rights for people working that land, as their labour becomes more valuable. People sometimes wonder what exactly happened to all those laves the Roman Empire had. They didn't disappear. Instead, their labour became more important and thus they were given more rights. This is something that began even before the fall of the Empire; as the slave population stagnated, slaves were given more protections against abuse by their masters, a logical step to protect the economy. On the other hand, the labour requirements also led to laws designed to reduce labour mobiliity, meaning that our self-made man from Africa was no longer really possible.

So, in the medieval period, that average farm worker would have had more rights than a slave, but less than a free worker. They likely would have had more food available, but less security against the privations of raiding peoples. When it comes to a subjective question like 'quality of life' it's not possible to say with certainty who had it better, since it depends on what one places importance on. However, I'd say that having more food and more rights trumps the small chance of a raid. This equation would shift in areas where such raids were more prevalent and stability was lower, and might lead me to say that Ancient people's had it better. All that said, the average life for either person isn't going to be that high quality compared to modern times.

If you're looking for additional reading, I suggest looking around for analysis on the pre and post Black Death period, particularly it's effect on labour, to see how the fall of Rome might have benefited the average person.

There are a lot of posts here that will possibly include answers to your question