Learn Finnish: Me opimme uusia tekniikoita toinen toiseltamme tatamilla. - We learn new techniques from each other on the tatami.

This is the formal distributive structure for reciprocal pronouns. The first toinen is in the basic nominative case. The second toinen takes the case required by the verb (here, the ablative -lta for learning from) and adds the possessive suffix matching the subject me (which is -mme).

Why does the second toinen specifically use the ablative case (toiselta-)?

The verb oppia (to learn) requires the ablative case (-lta or -ltä) when expressing the person you are learning from. Because of this verb rection, "from each other" becomes toinen toiseltamme.

Is it strictly necessary to use the double toinen structure, or could I just say toisiltamme?

You can absolutely use toisiltamme. In spoken and everyday Finnish, the single plural word toisiltamme is much more common. The toinen toiseltamme structure shown here is more formal and emphasizes the individual, one-on-one nature of the exchange.

Why is uusia tekniikoita in the partitive plural rather than the accusative?

It is the object of the verb oppia. Because learning techniques is an open-ended, continuous process here, and the exact number of techniques is unspecified, the object must take the partitive plural.