He mentioned how a person didn’t want to share their family online as someone might “steal it”. He then went on to say that since that person had compiled their family history from publicly accessible records, he didn’t see the point in hiding it.
That’s true to a certain extent. The information is out there, if you take the time to look for it. But, your family tree is the result of looking at original records, and interpreting the data to decide how all the family members fit together. That is your work, your time and effort. What I think the person was trying to say is that they didn’t want to share their information online as they didn’t want someone else to come along and copy all that information, without taking the time to verify the information against the original records. There are many errors in online family trees because of this issue, so it is a valid concern.
On the other hand, those new to family history can sign up to ancestry.com, and merge records into their tree, without checking to see if it even made sense. The fact is, there will be always be errors in online trees, whether a person copies someone else’s tree, or simply doesn’t think things through before adding records to their tree. I therefore agree that it is better to add a well-documented tree to the internet, so that others can advise you of any errors you have made, and you can provide someone else distantly related to you with information to help them find the records to complete their own family tree.