The greatest part about being a priest is being with people and families at the critical moments of their lives, and bringing the sacramental presence of Christ to bear on whatever situation they are in. Whether it is offering the Holy Sacrifice of the Mass, or annointing the sick for healing, it is a great blessing to be a shepherd of souls, and to continue the earthly ministry of Christ and His Church until He comes again.
Hearing confessions has been one of the most incredible experiences of God's grace and mercy. Many confessions I have heard have been a first confession, or from someone who hasn't been in years. It has been great to recommend the sacrament, and teach people about its necessity and place in the Christian life. I have found that most people do not mind making their confession at all. They may not go a lot, or maybe have never even been before, but when you ask them if they want to confess anything privately, and explain to them how it works and what it is all about, they gladly make their confession, and then are overwhelmed at the reconciling love they experience as they encounter Jesus in this beautiful sacrament. I firmly believe in the importance of regular confession in the Christian life.
I don't know how regularly other priests hear confessions (and I do not think priests should hear confessions if they are not regular penitents themselves), but if you do not, and would like to minister to people in this way, I would just make sure you ask people. Like a lot of other things in life, you have to ask people if they want to do it. They might say no, or have nothing to confess at that time, but at least you gave them the opportunity, and they know the offer is always there. But we have to ask. When you sense a moment coming on, where they are pouring their hearts out, why not gently recommend that it be done in the context of the confessional? It is like sales: you can have agreat presentation and product and so on, but if you never get to that point and ask, "Okay, so will you sign the contract?" You might miss out on a lot of sales. I do not like "sales" analogies very much, but it gets my point across. There is so much that we do not have simply because we do not ask.