In regards to the past few comments (sorry that I’m not quoting them all): In my experience, there are a number of things that matter:
1. Athletic history: your average wrestler starts out as a kid, and spends his teenage years hauling people in his weight class around, doing calisthenics, tumbling etc. He will build a pretty reasonable all-round athletic base that way. Now, if somebody starts out in a combat sport in their twenties, thirties or later after a much less active youth, things are definitely different, and you will definitely have to build an athletic base while learning the new activity. A number of approaches can be suitable for this. What I personally have found is that most people tend to focus a bit much on purely “fitness” type activities and neglect others. I especially recommend learning simple tumbling (not necessarily fancy stuff, but being able to roll in a variety of ways is extremely helpful to the learning process; I usually recommend working through levels 1-10 of L.L. McClow’s Tumbling Illustrated or taking a tumbling class for adults).
2. Carry-over between various athletic qualities: Athletic qualities are not isolated, they interact. I would especially like to point out the reciprocal relation of strength endurance and strength. It has been stated often that maximal strength carries over to strength endurance, but the reverse is also true. Most people will increase their strength also through mat work, which is arguably more strength-endurance than pure strength. Depending on a number of factors (fiber distribution etc.), some people may have a bigger carry-over in one way than in the other. I personally usually had more success increasing my maximum strength through strength endurance work than vice-versa, which meant I was increasing my work capacity at the same time as my strength, and I also found it easier on my joints. However, your mileage may vary.
3. Style: Wrestling is more focused on stand-up work, and generally a more athletic activity compared to BJJ. Therefore, the requirements for both leg strength and tumbling ability are higher. However, pulling strength is required universally, although the ideal choice of pulling exercises may vary from style to style (gi, no gi, favored attacks…).
4. Specific strength: If we are talking about strength standards for grappling, it stands to reason that the reality of the sport should be taken into account. If we leave out the specific requirements of parterre in wrestling, it comes down more to bent-arm pulling variations and squat variations (more so if takedowns are prominent in the style). These are the exercises listed in the wrestling strength standards above. Now, the level there is admittedly quite high (pull-ups with 70 kg additional weight for a middleweight, half squats with triple bodyweight…), but they can be scaled down. In particular, the seal row is a very useful exercise for all grapplers. I should also point out that the standard includes strength endurance as well, using the same exercises for the most part.
If I had to give a very simplistic approach to testing grappling strength / strength endurance, I would refer to two standards of the Russian Sambo team: rope climbing and squats with a partner of the same weight class (held in fireman’s carry) for reps. According to Alexander Iatskevich, the pulling standard of the Soviet Judo team consisted in climbing a five meter rope five times in a row without using the legs regardless of the weight class. I like this standard for a number of reasons. First off, the direction of pulling in rope climbing is to the center of gravity (the navel), which is the most common direction of pulling in most grappling styles. Plus, you have instability, unilateral pulling and the sequence of reaching, gripping tightly and pulling, both of which happen all the time in grappling. I would argue that in order to meet the standard, a person will develop a maximum of at least 20-30 pull-ups in a row on the side, and will be able to do at least ten reps with some weight (15-20 kg) added, which will translate to a one-rep max of 35-50 kg added weight. As for the partner squats, the standard I found is ideally 40 (!) full reps in a row. What that will mean for your one-rep maximum will again depend, but I would argue it should at least be in the ballpark of double bodyweight. As it has been written before, that will mean that the athlete should also be able to deadlift at least double bodyweight.
These standards may seem challenging both in terms of intensity and logistics, but again they can be scaled down. As for logistics, a standard wrestling or judo hall should offer all the required things, including a variety of partners to lift. In fact, some of the schools I have trained in recommend training strength right at the end of a mat session. That way, you will be the most recovered from strength work when you step on the mat the next time. A very simple way of doing some extra strength work is simply 1-2 sets of partner squats and 1-2 sets of rope climbing, peg board climbing or pull-ups at the end of your mat session, possibly also 1-2 sets of push-ups (simple push-ups, divebombers or one-arm push-ups). That only takes 5-10 minutes of time (as you are already warmed up) but can yield significant gains over time. I personally made a habit of arriving half an hour early for some extra solo drills, and staying fifteen minutes longer for some more strength work. If that is feasible in the academy you guys are training in, I highly recommend giving it a try. Add some easy roadwork in the morning or on off-days, and your conditioning base should be set.