Is it good idea to allow users to use remote image url as Avatar?

I am working on online community where users will have the profile page , where he can upload the image of his choice or give the url of the remote image .

So Is it good to just store the remote image url and not the image itself and use it like this on profile page like this

<img src="remote_image_url">

or download the image from remote url and store it on the local/our server for the later use like this

<img src="path_on_our_server">

i am thinking from the hack-proffing point of view , as are there any issues if i allow users to use remote image and use it as it is instead of downloading it on to our servers ?