First you should get a solid contract written up by an IP attorney that will clear up all the ambiguities. That would include, at a minimum, a confidentiality clause and a clause in the contract that states that all IP produced by the freelancer is a work for hire and belongs to the employer (you).
The concept of a logo is not protected, if you want an American flag logo, or animal logo, you cannot reasonable prevent the freelancer from ever making a logo with an American flag or an animal in it, in the future. If they took the files they produced for you and just tweaked them a bit, that is different.
If you have already started, and do not have a contract, chances are the final product and the work in progress are considered a work for hire, but it will be more difficult for you to prove in a dispute.
You should contract an IP attorney in your area to advise you on all the details of protecting you IP.
As my colleague notes, all of your concerns can be alleviated by contract. Properly written, an "assignment" clause in your contract with the artist will transfer to you the copyright in the drafts and final design he creates. The agreement you need is an Design Agreement that memorializes all that the artist will do [create], all that you will do [pay] and the consequences [you own everything]. Speak with your own California-licensed intellectual property. And note that you do NOT need a "work for hire" agreement because a graphic design is not one of the nine types of works that Copyright Law designates can be a work for hire. Banish that term of art from your vocabulary. Good luck.
The above is general information ONLY and is not legal advice, does not form an attorney-client relationship, and should NOT be relied upon to take or refrain from taking any action. I am not your attorney. You should seek the advice of competent counsel before taking any action related to your inquiry.
I imagine a scenario where you are working with a designer (i.e. a creative person who makes their living off of creating) and together you play around with ideas (concepts) and finally the designer has created an idea you like. The designer provides you the idea in a final form that you can use; for your business cards, websites, etc.
Now you ask: do I own every idea we visited along the way? While you do (with the properly written agreement as Mr. Ballard described) own 100% of the final design you bought and paid for; the variations along the way were rejected by you, and in this type of creative environment it would be difficult to say that every idea a designer threw on the table for you to consider instantly became yours to claim ownership over.
It should be (but it isn't) a matter of fairness. You either hire a designer to create one output, the logo, or you hire the designer to give you X time of creative work, so the designer knows your intent is to own everything the designer creates during that time. At least under this second scenario, the designer will not accidentally use a design they thought you had rejected for another client.
There are a thousand scenarios we could review, but in your case, only one counts, and that is the scenario that actually happens. To answer copyright ownership issues over that scenario, you need to discuss in much more detail, and with a copyright attorney.
My disclaimer is simply that Avvo already has an adequate disclaimer.
What you may not understand is that often, there are two or more owners of a copyrights in designs. For example, in the world of music, there are often two songwriters who share ownership of copyright. So to in this situation, you and the freelance probably share ownership in the designs, including work in progress, concept variations, and the final product. In the absence of an agreement the freelancer would share ownership with you. However, business owners routinely avoid this problem by requiring freelancers to enter "work for hire" agreements pursuant to which all designs created during the course of the project are assigned to, and belong to the company (or whoever hired the freelancer), These agreements are not simple--don't even think about using a Legal Zoom form. You need to retain a lawyer to draft a proper agreement for you---if this agreement is not drafted with appropriate care and precision, it will not do the job. These can be messy and complex situations, and drafting an agreement that takes account all of the specific nuances in specific cases takes skill. patience and experience. You need one of the fine California lawyers who regularly contribute on Avvo to help you draft this agreement, and negotiate its terms with the Graphic designer. By the way---it is quite foolish to have started this work with the graphic designer before having an agreement in place. This can be fixed by agreement only if this freelancer cooperates. But you need to understand that most freelancers/independent contractors understand the law a lot better than you---and most of these folks insist on a higher amount of compensation for "work for hire" agreements than for agreements pursuant to which they retain ownership of their copyrights and retain the right to share in the proceeds from the works that they create over time.
The ownership of the copyright in the work will depend on the language of the contract. If there is no contract, then the copyright would be owned by the designer (and you would have an implied license to use the work).
Different people may draw different lines between what counts as a "concept" and the execution of the concept, but copyright law generally does not protect abstract ideas, so it's unlikely you would have any copyright protection in a concept. This is certainly true if it hasn't been written down or otherwise fixed in some tangible medium.
As for other IP rights, the entity that actually uses the logo in connection with goods and services will own any trademark rights that are developed.
I'd recommend having an IP attorney look over and perhaps revise the language of your contract with the designer.
I am an attorney, but I am not YOUR attorney. By providing free, generalized information, I am not entering into an attorney/client relationship with you, nor am I providing legal advice applicable to your particular needs.