In general, a permit is required for renovations on the building framing skeletons (support beams, columns...etc.), and change of building facet/footprint through expansion or reducing. No, interior works not involving the main support components/framing usually do not require permits (unless you are changing "the use" of the building that involves "zoning", but it is advisable to have an architect/engineer go over your plan to ensure the integrity of the building structure will not be compromised.
However, per the language of the Ohio Building Code cited below (article 105.1), the addition of an interior wall can be seen as "alter" a residential building or structure that would require the approval (permit) from the department of building, though not specifically listed as such. You do need to check with your local building authority or architect for clarification.
BTW, there are two types of walls in a building: 1) Load Bearing Wall. 2) Partitioning Wall (usually non-load bearing).
If a permit is required for such work, the permit-issuing agency will let you know the stages that inspection is required, but usually, it will be conducted upon completion to ensure the work follows the approved plan.
Excerpt from 2017 Ohio Building Code:
"102.10.2 Minor repairs. Minor repairs to structures may be to residential structures made without application or notice to the residential building official. Such repairs shall not include the cutting away of any wall, partition or portion thereof, the removal or cutting of any structural beam or load bearing support, or the removal or change of any required means of egress, or rearrangement of parts of a structure affecting the egress requirements; nor shall ordinary repairs include addition to, alteration of, replacement or relocation of any standpipe, water supply, sewer, drainage, drain leader, gas, soil, waste, vent or similar piping, electric wiring or mechanical or other work affecting public health or general safety.
105.1 Approvals required. Any owner or authorized agent who intends to
construct, enlarge, alter, repair, move, or change the occupancy of a residential building or structure, or portion thereof, or to erect, install, enlarge, alter, repair, remove, convert or replace any electrical, gas, mechanical, plumbing system, other residential building service equipment, or piping system the installation of which is regulated by this code, or to cause any such work to be done, shall first make application to the residential building official of a certified residential building department and obtain the required approval.
Note, IMO, the permit requirement for interior minor works (adding/removing non-load bearing walls) often are "non-enforceable", provides you are certain there is no negative consequence (collapse) that will result from the activities and they do not disturb your neighbors or have a neighbor hates you very much. An architect or engineer can assure you of the safety concerns, rather than the city (the City would require the sealed plan from an architect or engineer anyway, or purchase a standard plan from them if there is one for this case).