Generally speaking, you turn your own attention, but you draw (or invite) the attention of others. In fact, when protocol is very important, and you are the junior person, you should invite their attention.

If you are speaking on behalf of the group, you can turn your (collective) attention to the next topic.

A thing, all on its own, draws your attention. Like the bright red cardinal draws your attention to the bush.

You can also "invite a lot of unwelcome attention" through your actions, as well as deliberately. (Like wearing really sexy clothing or - eek! passing gas during a quiet moment in church.) (Philip, that was for you - with your enjoyment of my colorful examples. )

Let's say that you said something that was contrary to the view of everyone else. Later, I may say to you "You sure drew a lot of attention with that remark."