Android Share Via Dialog

This is indeed done with Intents.

For sharing an image, like in the example picture, it would be something like this:

Intent share = new Intent(Intent.ACTION_SEND);
share.setType("image/jpeg");

share.putExtra(Intent.EXTRA_STREAM,
  Uri.parse("file:///sdcard/DCIM/Camera/myPic.jpg"));

startActivity(Intent.createChooser(share, "Share Image"));

For text you would use something like:

Intent share = new Intent(Intent.ACTION_SEND);
share.setType("text/plain");
share.putExtra(Intent.EXTRA_TEXT, "I'm being sent!!");
startActivity(Intent.createChooser(share, "Share Text"));