By definition, if a group says something offends them, then it is offensive.
By definition Native American mascots and names for sports teams are offensive. They are offensive because they offend Native Americans. The definition of the word offensive is that it offends people. It does, therefore, it is. Native Americans do not have to defend their offence or provide reasons for it. If a tall person doesn’t like the nickname ‘stretch’ because they have been made fun of for being tall all of their life, it doesn’t matter if you don’t think the nickname is offensive. The nickname becomes offensive because the individual finds it offensive. To argue in contrast to this, would be to assume that some offence is 'unjustified' and that there is a right and a wrong way to emotionally react to something. Making offence objective in this way raises the question: who is qualified to arbitrate? Who decides what emotional reactions to a statement or image are justified and which are not? The fact is that one person's offensive comment is another person's engaging debate and another person's self-evident truth. The offence is, therefore, entirely subjective and we must accept that if someone is offended, the comment or image becomes offensive.
If Native American communities were genuinely offended by the use of their imagery in sports teams’ names and mascots then it would be offensive. But most of the people expressing offence are not Native American. In fact, when polled, nine out of ten Native Americans have no issue with their depiction as mascots or the use of their names.  Just because other demographics say they should be offended doesn’t make the act offensive. If the actual community under “attack” isn’t offended, then the act isn’t offensive. Many high schools located on Native American reservations actually use the term ‘Redskin’ in the names of their sports teams. They also use Native American figures as mascots. If they are embracing the term and are not offended, then it is clearly not a problem.
[P1] If people are offended by something, then that thing is offensive. [P2] Native Americans are offended by their depiction in sports mascots and names. [P3] Therefore, their depiction in sports mascots and names is offensive.
[Rejecting P1] Things are only offensive if the people who are under 'attack' are offended. [Rejecting P2] Nine out of ten Native Americans have no issue with their depiction as mascots and the use of Native American names in sports. [Rejecting P3] Therefore, the act is not offensive.