It would help to mention your or your contractors country. Do we talk US or Japanese or German or French or Russian copyright law? Some countries have clauses that explicitly govern who holds copyright or if there is an exclusive license transfer in case of conracted/employed work ("Work for hire"), others might be silent. So I give two examples:
In the US, the concept of Work for Hire under 17 USC § 101 is defined in such a way that if somebody directly commissions a work (like, if I would pay an artist to add illustrations to a book I do) or you are employed to do some copyrightable work for them (for example, if you would be employed by Paramount Studios), then said commissioning entity/employer is vested with all (transferable) copyrights. This provision demands written contract of both parties, and for commissioned artists it needs to fit one of 9 groups, among them collaborative work (the illustrations) or explicitly work on audiovisual works like games or YouTube videos.
Under German law, the Urheberrechtsgesetz (~Law regulating authorship rights) defines that only a natural person has Urheberrechte (~authorship rights), which encompass a group of three things: Right to be indicated as an author, the right not to have your work defaced(~moral right) and the Verwertungsrecht (~usage rights). The first two are inalienable and the first one can't be transferred at all, while the moral right not to have the work defaced is only transferable via inheritance. And then comes the usage right. German law says, that if you are employed to do a piece of work, then the employer automatically gains the usage rights. However, here again, it helps to be contracted to make sure that the transfer of the exclusive usage license (as in: you retain no license to use the cut work yourself) is properly done.
As a result, it's always better to have a written contract to point to, preventing Hesaidshesaid and other interpretation problems. Often, such contracts contain language that works for more than one country, like referring not to copyright but "All transferable rights in the work" to make the contract enforceable in varying jurisdictions. Besides the safety it brings for the employer/contracting side, it also should give the employed/contracted side a document to point to which directly governs how much you are owed or what is to be taken as the basis for the billing rate.