I am considering making my investment history publicly available online
What is the benefit you are looking for by doing this? Just to establish that you are a successful investor, so in long run can predict things ... have tons of followers? If so yes. Go ahead. Updates to the portfolio would have to be near real-time than post facto else no one will believe you and it would be useless.
are there any reasons (legal, personal, etc.) not to publicize my personal investment history
legal, depends on country; I can't think any [check the agreement with your broker / depository] on how much can be displayed. i.e. they may forbid from revealing contract ref / or some other details.
On Personal front, it depends who takes a liking to your stuff.
Relatives: They know you are making huge profits and may want to borrow stuff ... or queue up to you requesting to make similar huge profits for them; only to realize when there is loss they blame you ... this can strain relationships.
Friends: Although close friends may have a general idea, if you are too successful and it shows; it can have its own set of issues to deal with.
Colleagues / Manager: If you are too successful, it may mean you may notionally be earning more than them ... they would start unconsciously monitoring your behaviour ... this guy spends all day in office researching for stocks and doesn't work. That way he knows how to pick good stock ... he is wasting company time. The same happens if you are loosing stock ... a unrelated bad day you are having maybe equated to loss in stocks. Depending on the job / roles, they may move you to different role as the perceived risk of you swindling goes up.
Generally important work doesn't get assigned, as it would be assumed that if you are successful in investing, you may quite soon and start full time into it.
Identify Theft: As mentioned by keshlam, to much data one can easily risk identity theft. Realize phone banking to get some routine stuff just asks for basic details [that are available on face book] and few recent debits / credits to the account. This will be easy see the trades you have done. None of us here are expert identity theifs. But the real one have tons of way t