Immigration into the Empire is a huge topic, both from the cultural (what impact did migrants have on roman life?) as well as the military (how were the borders controlled?) perspective. Rome lasted a thousand years, and for half that time it had a gigantic border, so it is difficult to generalise an answer. Most often, actual frontier policy was up to the local magistrate, but appeals could be made higher up the hierarchy up to the emperor himself.

In general, roman cities themselves were extremely diverse and attracted immigrants from all over the empire. While today we might not consider these people 'immigrants' because they originated from within imperial borders, for a Roman a provincial rather than urban background is a much more important distinction. Roman attitudes towards these foreigners are as diverse as ours today, from 'egyptomania' or 'germanic subculture' to xenofobia and conservatism. But I think that is not what you are primarily interested in.

So let's have a look at actual frontier policy. From our written sources we get the image of an extremely restricted border, with many cases of people requesting asylum and instead being massacred or enslaved, and at other times people being kept in concentration camp-like circumstances, being starved to death at the border. Yet at other times asylum was granted, even to entire armed groups with large amounts of autonomy within the empire. In this sense, Roman attitudes seem to be entirely dictated by realpolitik, with little concern or morality for people themselves. But then again, this situation of client states requesting alliance is entirely different from the destitute and solitary refugee, or the adventurous individual trying to make a new life. These are armed polities trying to elevate their social status, so obviously they are dealt with politically.

Archaeology is telling us a much more nuanced story of cross-border migration. From the many Roman artefacts found north of the border it is evident that many people who once had employment within the empire, must at some point in their life have returned, bringing Roman habits and styles with them. This might even have included Roman craftsmen, architects, or advisors, so the movement of people was very much two-way. The most visible profession we can identify is obviously that of the soldier, with serving for some time in the Roman army being a relatively common step in the career of men outside of the border. But there is no reason to assume other forms of labour migration did not take place as well. After all, the primary architectural contribution of the empire at the frontiers was the building of roads and bridges, not of walls.

It should also be considered that the Romans did not see a the Rhine or Danube as a frontier in the sense that we do today. Roman ideology emphasises that Rome is an empire without frontiers, and the limes was just that, a temporary line within Roman territory. Romans would not necessarily see a difference between barbarians within or without that line. This changes somewhat over the course of the empire, and as the limes becomes more permanent. It is only about the fourth century or so that there is an increasing awareness of 'us vs them', further enhanced by the conversion of the empire (and later of the arianism of the barbarians).