The way the gooseneck flushing flow path in the toilet is made, the water in the bowl stops dropping at the elevation of the bottom surface of the top of the gooseneck passage in the toilet - you can usually see the gooseneck path on the outside of the toilet as a raised curved surface. This raised portion of the gooseneck passage prevents the toilet bowl from completely draining out when it flushes, thereby forming a trap to keep sewer gases out. The only way the water in the bowl can stay noticeably above that level is if there is a blockage, regardless of how much water the tank may be leaking down. Case in point - when a toilet keeps "running" because of a flapper leak or stuck fill valve, the bowl does not overflow unless it is blocked - the water just endlessly runs into the toilet as a slow stream and out the drain and down through the gooseneck, maintaining the same level in the bowl - sometimes to the tune of thousands of gallons per month.
This is the level the water wants to reach - normally it flushes down rapidly and essentially empties the bowl, then refills from the fill valve in the tank to a low static level. If the bowl is overfilling during the flush (near to the rim) or staying at a higher level after the flush, then you have a partial blockage, likely in the gooseneck but remotely possibly in the drain pipe between the toilet and where its drain pipe meets the one from the other toilet backed up to it.
My recommendation - a plunger may work but may not be needed. If it is caused by paper or some foreign material draped over the top of the gooseneck but not fully blocking flow, or a floating item like a child's toy, frequently using a bucket with full hot water from the tub/shower poured down it will flush it out. Pour in as fast as you can to fill the bowl to the underside of the rim without splashing, but don't let it overflow of course. Mix a bit of dishwashing or liquid (not oily) hand soap into it to help lubricate the process. Usually a couple of buckets full of full hot water will do the job, as hot water breaks down toilet paper pretty fast. If not, then plunger time.
One thing with plungers - the blockage got into place under the force of the flushing water, so it is stuck in the "forward" direction. The bulb or piston type plunger that was in a prior comment pushes forward, so if it does not push it clear, it can cause a plug to tighten up and become a full blockage that will then have to be snaked. To unplug a blockage, I recommend best way (if hot water will not work) is to use a conventional plunger like this -
that fits over the entire hole in the bottom of the bowl, GENTLY compress it so the center is depressed well into the cup, then release pressure quickly so it pulls a suction on the gooseneck, pulling the blockage back towards you. Sometimes, with partial blockages caused by toilet paper "plastered" in the drain, a back and forth surging action works best - but be careful about bubbling and splashes.
For a difficult blockage, then you need a toilet snake or closet toilet auger (a snake with protective cover so the wire snake does not scratch the porcelain) - looks like this -
cost about $20-50 depending on brand and length, or call a plumber or sewer and drain contractor (usually a bit cheaper) for about $75-150 service call charge.
One comment - a LOT of blockages are being caused by the multi-ply quilted toilet papers - my favorites plumbers say Quilted Northern and Quilted Charmin are among the worst, because they break down slowly. My favorite sewer and drain contractor also blames them on a significant increase in calls for plugged sewers and leach fields - he sees a lot of sewer pipes with a heavy matted lining of toilet paper sticking to the grease lining the pipe, reducing the diameter of the pipe. Both say that in recent years since the quilted paper became popular the incidence of slow draining toilets versus blocked has gone up significantly. Odds that that is the type of paper you use ?