GH does shut down your natural GH secretion, but it's only for a few hours.. GH is naturally released throughout the day in a pulsatile fashion. About every four hours you'll have a GH release from the pituitary, after which point, GH's negative feedback loop kicks in in response to GH and IGF-1 plasma levels to upregulate somatostatin. This occurs in roughly 4 hour cycles throughout.....life. The amount that is secreted at each pulse varies with age and also state of consciousness. You release much more GH during sleep than you do during the GH pulses during the day.
Taking exogenous GH elevates your plasma GH levels and incites the GH negative feedback loop to commence, so it'll throw off your normal pulsatile release, but this will just begin again roughly 4 hours later.
This is why taking GH doesn't need to be everyday. It's effects are carried out within hours via GH itself and IGF-1. The reason we like to use it for a continuous period is that the cummulative results are easier to see.