That depends, if the parties are listed jointly on the home title there would be an equal right to share in the property. However, if only one person is on title, the other individual will have no presumptive right to share in that property.
So, the personal title can ask the other person leave the property if they choose?
Yes, only legally married individuals have a right of possession which requires a court order to be forcibly removed, common law spouses do not have this right so the homeowner can deal with their property as they choose.
What if the person not untitled contributed substantially to the home during their time there?
That party should apply to the Court for recognition of those contributions, the court will then determine what interested, if any, that person has in the property.