It is true that anything the slave owned would have belonged to the master. Technically, the head of the family, the paterfamilias had the power over all the possessions of the people in his househould, under his potestas, excluding the wives' dowry (that is in a traditional manus-marriage, there were other forms of marriage that kept the possessions of both partners separate). So he was the sole legal owner of any and everything his wive, sons, daughters and slaves possessed until the time they left his power, since only persons who were sui iuris, of their own law, were able to own property. This is obviously a very unpracticable state of affairs.

The solution was not particular to slaves, but to all members of the familia who were under the power of the paterfamilias, and this was the concept of a peculium. In its original sense it is a special asset separate from other assets, in the common sense it meant those special assets that a paterfamilias entrusted to one of his subjects for them to work and manage, for example a small herd of animals, a building or plot of land and its profits, or a loan to run their own shop. De jure these assets are neither in the possession nor under the ownership of the holder of the peculium, their parts are still part of the assets of the owner. He or she only has detentio, that is the de facto possession and the ability to do with it what he wants. This has several advantages, for example it allowed sons to keep what they earned in the military (pecunium castrense) or through other service (quasi castrense); another advantage was that the master would not be liable with all his assets for the dealings that one of his slaves (or one of his profligate sons) undertook with the peculium entrusted to him, while still retaining possession of it; so a kind of ancient limited liability company, if you will. It also allowed the master to skim off the profits or dissolve the peculium at will. Since slaves were indispensable in the running of day-to-day business, this was an easy method to give them a freer hand, so they would need less control and could act on their own.

The biggest advantage in regard to slavery seems to have been that it provided a motivation for slaves to work hard for their master and profit from their own work. Varro in his treatise on agriculture, where slaves are discussed as part of the articulate tools needed to run a profitable farm, advises the farmer to give his slaves the opportunity to own some things of their own which they could profit from and use for their own goals.

They are made to take more interest in their work by being treated more liberally in respect either of food, or of more clothing, or of exemption from work, or of permission to graze some cattle of their own on the farm, or other things of this kind; so that, if some unusually heavy task is imposed, or punishment inflicted on them in some way, their loyalty and kindly feeling to the master may be restored by the consolation derived from such measures. [Varro, rust. 1.17, 7].

Having some cattle of their own to look after and profit from would have been a form of peculium fitting for a farm, and the word itself comes from the word for a small farm animal, pecus. So a slave could de facto have possessions of his own, even if they de jure belonged to his master, but there was a strong moral and economic obligation to honour this peculiar state of affairs, so it became common practice. I don't want to overstress the analogy, but in a certain sense it allowed the slave to become a share-holder in the masters enterprises, tying his economic success to that of his master. It seems obvious that this was a big motivation for the slave that was so minded to do his work well and look after his masters profits that thus became his own, in this limited way.

It also seems that the peculium normally stayed with the slave when he was transferred to a new master:

In the purchase of slaves, it is customary for the peculium to go with the slave [Varro, rust. 2.10, 5]

Again, the important word here is customary, it was all up to the disposition of the master. He retained full ownership and could withdraw the peculium for any reason and at any time should he be so inclined, only facing the obstacle of overcoming common practices and mores, damaging his public image (which should not be underestimated in a society where people were so preoccupied with precisely that). What I want to say by this is that your mileage may vary, and if you are a slave on some far away plantation or any other remote place, who are you going to look for to complain?

But, should the slave be freed at some point, he would also have retained possession and gained ownership of his allowance. In short, while a slave could not legally own anything, same as any other member of a household under the pater, there was de facto a possibillity to own and spend (and profit from) anything ranging from some pocket money or a small flock of sheep to a whole business operation. When Musicus Surranus, a slave of Tiberius, died some time in the 1st century, he was in possession of no less than sixteen under-slaves, or vicarii [see inscr. CIL VI 5197]. Slaves owning other slaves was another instrument to encourage loyalty and productiveness, and also competition among the slaves, something that might seem strange at first but makes perfect sense considering the things said above.

Now, as regarding buying their freedom, the evidence seems a bit mixed. It is certainly true that slaves could buy their freedom, like Publius Decimius Eros Mentula, who bought his for 50,000 sesterces [CIL XI 5400] and became a great and wealthy benefactor of his home town after this. But Mouritsen has recently argued [H. Mouritsen, The Freedman in the Roman World (Cambridge 2011), see here esp. p. 177ff.] that this was less often the case than is commonly assumed. One of his arguments is that there is less literary evidence for it than we like. The other, to me more convincing one is that it made less economic sense in the special context of Roman slavery.

It is true, buying your own freedom is a common staple of slaveholder societies ancient and modern, and a common trope, the deserving slave buying his own life back after hard work. However, the Roman freedman after his manumission still was in a very deep relationship with his former master, who now became his patron. Upon gaining his freedom, the slave took on the family name of his former master and thus became a full member of the masters family (and de facto this was one of the ways to perpetuate a family). One does only need to look at the way the Tulli Cicerones treated and cared for Ciceros slave and later freedman Tiro. He truly was family to them. Cicero and his brother Quintus in their letters show great concern both for Tiros health and well-being, but also for his professional success.

What this means is that, while the freedman still has obligations towards his patron, usually staying in his household and continuing to work for his former master in some capacity or another, the patron also has obligations to his client, to care for him and help him in times of need. Often freedmen would carry out their masters business interest, act as his authorized officers or trusted messengers and manage workshops or farms for him. Being the freedman of an important patron was an important position in and of itself, and the imperial freedmen running the emperors bureaucracy became a common trope in the 1st century. So, while it might not necessarily have served to buy his freedom directly, the peculium was the proof of a deserving slave, one who managed his own (and thus his master's) affairs well and worked hard, thus he was deserving of freedom. This might be where the connection between the peculium and manumission stems from. Simply taking back that peculium from the slave upon manumission would undermine that special 'social and moral bond' [Mouritsen 2011, 180] that would form the basis of their continued and important relationship, in which trust would play a very important part.

One example of a type of slaves where the buying of ones own freedom was more common were actors and doctors (like Publius Decimius mentioned above), people who could make their own living with the specialist skills they had aquired without the support of their patron. Others might have been old or infirm slaves, who thus financed their replacement in a perverse twist of business acumen. But generally, other forms of manumission (by testament or for their hard service, or simply magnamity) seem to have been more common.

Edit: Added some stuff and fixed some typos/smoothed out the language a bit.