How do Catholics reconcile these verses about giving publicly and yet privately? I understand that the Bible encourages charity, however my question is in regards to how it proposes we ought to go about performing charity according to Catholic Christians.


  "Do not repay anyone evil for evil. Be careful to do what is right in the eyes of everybody."  - Romans 12:17 NIV


This verse would seem to suggest that one's charity should be well known to others. The context of this verse would suggest to me that others should know of our kindness so that they have a favorable view of the church as a whole.


  “Be careful not to practice your righteousness in front of others to be seen by them. If you do, you will have no reward from your Father in heaven.
  2 “So when you give to the needy, do not announce it with trumpets, as the hypocrites do in the synagogues and on the streets, to be honored by others. Truly I tell you, they have received their reward in full. 3 But when you give to the needy, do not let your left hand know what your right hand is doing, 4 so that your giving may be in secret. Then your Father, who sees what is done in secret, will reward you." - Matthew 6:1-4 NIV


This verse would suggest to me that we ought to give privately so that our motivations for charity aren't tainted by pride or selfishness.

How does the Catholic church reconcile these two views?
The question is based on a misreading of Romans 12:17. Paul is not exhorting one to do what is right and to do that in sight of everyone. Paul is saying that one should do what everyone deems right: "What is right in the eyes of everyone, be careful to do that."

Consequently, it's perfectly possible to do what is right and do it in secret. There is no contradiction.