The general procedure for mechiras chameitz is as follows: you tell your local Rav to sell your chameitz to a non-Jew in the city, and he does so. Is there any reason why I can't sell my chameitz to a non-Jew, such as a neighbor, myself?
There is no Halachic restriction stopping you from selling your chametz to a gentile, provided that you make a valid sale in accordance with all the pertaining halachot. However this becomes a bit complicated once you take into account that you most probably will buy back the chametz from the gentile after pesach, this can potentially render your sale invalid if you don't do it properly and with full sincerity. In addition there is a timing issue, e.g. if you sell to early and you are still using the chametz, or if you sell too late and there was already a chiyuv to burn it, etc. For these reasons (in most cases) it is better to sell through your LOR.
As a side note I will mention a story I once heard from my Rebbi: In a particular kehila they had written in the mechira contract that they are selling all wheat products. They had mistakenly failed to write all leavened wheat products, thereby selling their matza together with their chometz & potentially being mevatel their mitzvah of eating matzah on Leil Pesach or even being mevatel the mechira (because they had no intention to sell their matza so when they signed the contract they were not actually having in mind to be bound by the language it used.). I don't remember when my Rebbi actually caught the mistake. This story goes to show how easy it is to botch up and cause serious issues.