The person in the right seat is not the SOE. SOE stands for "Supervised Operating Experience", which is simply the event you are subject to. I would say that you cannot log this as PIC based upon 61.64(4) which is the SOE rule as pasted below, unless you are logging it as sole manipulator. You are not the actual PIC though, as the regulation clearly states.
In blue, you can see your limitation should clearly state that you may not act as Pilot in Command until meeting the requirements in red, which is the Supervised Operating Experience (SOE) with a current and qualified person who is the PIC. You must fly from the normal PIC seat (Left in most cases) and the PIC must attest to the legs in your logbook.
(4) If the applicant does not meet the requirements of paragraph (a)(2) or (a)(3) of this section as appropriate, then—
(i) The applicant must complete the following tasks on the practical test in the airplane of the category, class, and type of airplane rating (if a type rating is applicable) for which the airplane rating applies: preflight inspection, normal takeoff, normal instrument landing system approach, missed approach, and normal landing; or
(ii) The applicant's pilot certificate will be issued with a limitation that states:
“The [name the category, class, and type of airplane rating (if a type rating is applicable)] is subject to additional pilot in command limitations,” and the applicant is restricted from serving as pilot in command in that category, class, and type of airplane rating (if a type rating is applicable).
(iii) The limitation described under paragraph (a)(4)(ii) of this section may be removed from the applicant's pilot certificate if the applicant—
(A) Logs 25 hours of flight time in the category and class of airplane for the rating sought, and if a type rating is being sought, the flight time must be performed in the same type of airplane for the type rating sought;
(B) Performs 25 hours of flight time under the direct observation of the pilot in command who holds the appropriate airplane category, class, and type rating, without limitations, in the same category, class, and type of airplane rating, if a type rating is applicable;
(C) Logs each flight and the pilot in command who observed the flight attests to each flight;
(D) Obtains the flight time while in the pilot in command seat of the appropriate airplane category, class, and type, if a type rating is appropriate; and
(E) Has an Examiner review the pilot logbook and endorse that logbook, attesting to compliance with the required supervised operating experience.