I am building a shed and want to side it in plywood that is 5/8th in thick with routed board effect, 4x8 sheets. I want to run the panels horizontally, not vertically. There is an overlapping joint (not tongue and groove). Do I need to put anything between the sheets of plywood where they meet. Flashing? Glue? Caulking? What kind of fastener?

You are opening yourself up to a bevy of problems by running the sheets sideways.

Yes,you would need a z-channel between the lower and upper sheets.but that's just the half of it.

The sheets would be sideways,making the end grains on an up and down plane.How to make a 100% seal there is beyond me,the expansion and contraction of any wood would make this difficult at best.The end grains are where moisture wicks into the sheeting the most.

Length of the building,how many overlaps.all play a part in how difficult this could all be.

Fastening any exterior plywood,I use twist nails(siding section, galvanized) or boxed siding nails(galvanized)

You need rabbit joints. You need a rabbit on the front side of one end and one side. You need a rabbit on the back side of the other end and the other side. This will give overlapping joints. Rabbits should be 5/16" deep (half the thickness) and 3/4" wide All joints, including the horizontal ones, must fall on a stud. Silicone all joints. Nail with galvanized finish nails.

Because you are in effect shorting the 4x8 plywood by 3/4", you must allow for this when placing studs.

KBC is correct.

Sounds like you have T1-11 siding which is not designed to be run horizontally.

Either run it vertically or you will have nothing but problems.