This is a very interesting question. I've gone through something similar myself and it irked me to no end.
As a DM student you are paying to take a class. If they offer to teach the class, and take your money, they should either state up front that there are certain gear requirements, or they should let you dive the gear you already own. Since you will be seen as a rep for their store during your internships, it would not be unreasonable for them to require you to use their gear, *so long as they state that upfront when you sign up for your dm class*.
Once you've finished DM, you are volunteering to work with the store to both of your mutual benefit. They don't necessarily owe you a thing, nor vice versa. If either of you wants to make stipulations, you should feel free to do so, but the other should feel free to terminate the relationship. (ie: LDS tells you to use SP gear, you tell them no thanks and walk away).
My case was particularly pertubing in that the LDS specified that they didn't want me to be seen wearing gear they don't sell even on my own time, nor to be seen walking into the store that sells said gear (the Halcyon dealer / competitor). I explained that I would do what I want on my own time, and we ended the discussion on friendly, if guarded footing. Since then I've reduced the number of classes I assist that store with and begun spending more time at the other (friendlier) store.
Were I to try to make my living selling dive gear I would keep a set of rental gear that I reserved for instructor/dm use. I'd ask that as a courtesy to me my staff only wear products that I carry when working with my students, and I'd welcome them to use the rental set as needed. I would also offer substantial discounts to the staff to help them acquire the gear that I sell. Whether paid or not, the staff are acting as part of the business and should act in the best interest of the business.