The things you have circled in red are not player arrows. They are point-of-interest arrows, or POI arrows, pointing to things that are not currently on the map, and are probably created by an addon, in which case you can hide them by turning off that addon, or configuring that addon to not show them. Post a list of your addons if you need help figuring out which addon the arrows belong to.
The player arrow is the arrow in the exact center of the minimap, indicating the direction you are facing. I'm fairly certain it cannot be removed.
They look like the arrows TomTom adds for waypoints... I can't remember the last time I was in a group with people who were spread out over a large area (like a battleground) though, so you're probably right, in which case turning them off is either impossible or more headache than it's worth.