They do give up some parts of a normal life. But poverty is not part of the calling of secular priests. They are not expected to give up all of their own money and possessions to the parish (as was suggested in one of the posts above) or to give up any idea of having fun. If a priest wants to have a fun car (or motorcycle) and has his own money for it, good for him.
This is rather sticky point. A secular priest does not make a vow of poverty. This is true. However, it is not true that secular priests are free to do as they wish with their money. This depends on where they are. A secular priest that lives in community is bound by the constitutions of that community (ie. Maryknoll, Vincentians, Missionhurst, SOLT, FSSP, ICRSS). These institutes do impose restrictions of the use of money, even though these priests are paid a salary. Vincentians and Maryknoll have very strict norms on poverty.
A diocese can impose restrictions on how its priests use money as well. Canon law says that a secular priest must always be a model of simplicity and humility. It is the bishop who defines what is simplicity.
Beyond this, it is also true that the pope, because he has universal jurisdiction can dictate to any priest or to any religious the parameters for the exercise of poverty. One of the points that has already been noted by superiors is that the Pope said to the seminarians and religious novices, “You can’t do this.” His message was pretty obvious. In a nutshell, he was telling us that he disapproves. As the superior, whatever he disapproves must be taken seriously.
I found it especially interesting that the descriptions that the pope was giving of simplicity and poverty are paraphrases from the Franciscan statutes. He very much reminds me of Pope Leo XIII who also tried to inspire the entire Church to adopt much of the Franciscan asceticism. I believe that it was Pope Pius XII who directed the bishops to promote Franciscan spirituality among the lay faithful. I would not be surprised if without ever mentioning the term Franciscan, Pope Francis continues to steer the entire Church toward a more Franciscan expression of the Gospel.
What is totally fascinating for anyone who knows the Franciscan and Jesuit schools is that he is promoting very Franciscan ideals using very Jesuit methods. If one pays close attention, one can see how the two schools complement each other and serve the needs of the Church, which is what the Holy Spirit intended when he raised them up. Providence decided that the Church needs a faithful religious at the top at this time in history.