You will need the Limited Write People permission to perform this action. However, you are able to change the image for your own profile if you do not have this permission.
Navigate to the person's profile and click on "update image" under their current profile image on the right.
A file dialog box will appear asking you to select the image to upload. It is recommended you update a square image. If the image is not square, you will be given the opportunity to crop it before uploading.