Am I required to buy the new chapters in order to continue to play Guild Wars?
No. Every purchase you make in the continuation of the Guild Wars saga will be your choice. If you purchase new chapters, you will gain access to new regions of the world, new skills and abilities, new items, new professions, and much more. And of course such a purchase will support the continued development of the game. However, if you choose not to purchase a chapter, you will still be able to play the chapters of Guild Wars that you own, and you will have common areas in which you will be able to play with and against your friends who have purchased the other chapter(s).
Will I remain competitive if I do not buy the chapters? Will I be able to compete with and against others if I have only some of the Guild Wars chapters?
Yes. Purchasing the newer chapters of Guild Wars will not make you strictly more powerful. You will have access to many more strategic options, due to the expanding nature of the skills, abilities, items and professions that you enjoy with each chapter. It would be similar to building a deck in Magic: The Gathering: The more cards you own, the greater the number of different playing decks you can choose to play. When you buy the chapters of Guild Wars, you will acquire a larger collection of skills and abilities from which to build your skill set, but you will not gain more power. So if you purchase a chapter and your friend does not, you will still be able to play competitively against and with one another.