Firstly we are not the best conductors, so current might be having a relatively hard time getting through us.
But I believe the real reason is that you also need a high potential difference in order to get current flowing through you.
Like lightning which needs a huge potential difference between the clouds and earth (so big that most of times a neutral earth does not give this high difference but needs some accumulation of opposite charge from that on the cloud).So without that potential difference nothing happens.
(Note: the Earth actually causes voltage difference at about 200V per meter, so we do have that big potential difference, but Earth tends to make every object that it is touching neutral--to have the same potential as its surface-- so we distort the equipotential lines that the earth produces. For more on this, check Feynman Lectures Vol2)