We are the professionals most responsible for determining who needs Special Education Supports and what kind. That's the shortest answer and if it works for you then, I won't be offended if you stop reading right now. It's also interesting to learn about how we do this, though.
One important thing to keep in mind is that we are part of a team. We take reports from teachers including class work, grades, their observations and formal measures such as ratings scales on which teachers report behavior and their responses are compared to a large sample of such reports. We do the same with parents and with other professionals (Speech and Language Pathologists, School Nurses, Physicians etc) who have worked with the child.
All of the reports and our observations and interactions with the child then form a big part of the picture of why a child might be having difficulty in school and help us interpret tests that we give. Testing includes Intelligence (including sub-abilities like working memory and reasoning) achievement and sometimes other areas such as attention. These give us a feel for where a child is performing in all of these areas under what are basically the ideal conditions since they are administered one-on-one with us there to encourage the child and assure that they are doing their best, aren't distracted and get a lot of encouragement.
Next, we put all of the information together and start to form a conclusion about why the child is struggling. For example, if little Timmy has had an uneventful upbringing, gets along well with teachers and peers and has typical intelligence (with maybe an area of weakness here or there) but he has great difficulty in reading, then it looks like he has a learning disability in reading. If little Julia can perform well on all of the testing with me, but she cries and refuses to do math in her classroom or with her parents, then maybe she's anxious about math.
Finally, we work with parents and teachers to develop a plan for how to help the child. In Timmy's case, maybe we recommend reading help and the resources that are available in our schools (such as reading specialist or small group instruction). Until his reading catches up, (which it sometimes will and sometimes won't) we may look at resources such as software that reads his textbook to him out loud. Depending on the level of care needed, we and the rest of the staff working with the child will determine whether this rises to the level of a Disability that requires Specially Designed instruction (that is "Special Education").
It's also good to keep in mind that such a thorough evaluation is not always needed. If a child has low vision for example, Special Education may just be a matter of enlarging his or her reading and math materials. If a child only has difficulty with speech and communication, then an evaluation from a Speech and Language Pathologist may be the only thing required. Some schools also have significant "pre-referral" processes such as getting a child into extra math or reading help as soon as possible and carefully tracking his or her progress. In other words, that's our job generally, but the specifics always depend on the child and the school.