The Current State
Most professional sporting bodies follow the Olympics' inclusivity guidelines. Currently, IOC (the International Olympics Committee) follows the 2015 guidelines for Transgender inclusivity, which has several key requirements for athletes to compete in their preferred gender bracket:
- Their gender must be legally recognised by their home country (excepting countries in which transgenderism is illegal)
- Transfem atheletes must have declared their gender publicly, and not changed it for a minimum of 4 years, and must demonstrate a testosterone level of less than 10 nanomoles per liter "for at least one year prior to competition, and throughout the period of eligibility."
- Transmasc athletes must have declared their gender publicly.
These rules were first applied to the 2016 Olympics, although more restrictive policies have been in effect since the 2004 Olympics.
The main issue cited when people take issue with transfems competing in women's sports is the greater level of testosterone that pre-HRT trans women possess, which leads to different muscle and fat distributions. However, a 2017 metastudy into the advantages of trans women found that "there is no direct or consistent research suggesting transgender female individuals (or male individuals) have an athletic advantage at any stage of their transition". In addition, the studies which have found differences between trans and cis athletes have noted that the estrogen supplements and testosterone blockers used by transgender athletes lead to a decrease in muscle mass and red blood cell count. The effect of this is that even if transgender athletes have greater bone and muscle mass, HRT treatments significantly reduce the advantages these provide.
For this section, let's assume a pre-HRT transfem athlete, who has the testosterone levels, muscle and fat distribution of a male athlete in the same age range. While this may be an advantage in rugby, for example, the extra weight, muscle, and bone mass is a disadvantage in a more athletic sport such as gymnastics. As a result, there is no "one size fits all" solution to the percieved advantage of transgender women in sports. Each sport has different requirements, and policy must be built around not only the biology of the athletes, but also the demands of the sport.
What about... ?
Transmascs: In professional sport, transmascs simply need to declare their gender. HRT and legal recognition are not required, as they are percieved as having a disadvantage because of their naturally lower testosterone levels.
Intersex: DSD athletes have different rules in different fields, but the most notable (and in my opinion, sensible) restriction is that of World Athletics (Source 4):
DSD athletes will be subjected to specific rules if they have XY male chromosomes, testes rather than ovaries, have circulating testosterone within the typical male range (7.7 to 29.4 nmol/L), and are androgen-sensitive so that their body makes use of that testosterone. World Athletics requires that any such athlete must reduce their blood testosterone level to 5 nmol/L or lower for a six-month period before becoming eligible for track running events from 400 metres to the mile run in international competition, though World Athletics publicly remains open to extending this to other events based on new scientific study. 
1. Sykes H. Transsexual and Transgender Policies in Sport. Women in Sport and Physical Activity Journal. 2006;(1):3-13. doi:10.1123/wspaj.15.1.3
2. Harper J. Do transgender athletes have an edge? I sure don’t. - The Washington Post. Washington Post. Published April 1, 2015. Accessed May 5, 2021. https://www.washingtonpost.com/opinions/do-transgender-athletes-have-an-edge-i-sure-dont/2015/04/01/ccacb1da-c68e-11e4-b2a1-bed1aaea2816_story.html
The author, Joanna Harper, is a medical physicist at Providence Portland Medical Center in Oregon.
3. Edwards M, McCormack A, Lauder J, McVeigh S. Testing, hormones, hatred: What it’s like to compete as a transgender athlete - Hack - triple j. triple j. Published July 28, 2016. Accessed May 5, 2021. https://www.abc.net.au/triplej/programs/hack/transgender-athletes/7669902
4. Anderson E. Transgender Athletes in Competitive Sport. Taylor & Francis; 2018.