LSE accepts about 175 new Law students a year.
UCL accepts about 275.
So, rounded, LSE has 64% of the number of Law students that UCL has.
They're both quite small compared to some other universities. Bristol and Exeter accept 400 each, for example. KCL has 1048 at a time so maybe 350 new Law students a year.
The Chamberstudent report from 2019 (I know people don't like it because it's several years old and a limited sample and the proportion of students at each place might have notably changed since then) say that:
KCL provide 3.4% of Law trainees.
UCL provide 3.1%
LSE provide 2%
So, rounded, LSE were providing 65% of the Law trainees that UCL were. In other words, there was no real difference between the 2 universities for this, proportionately. However, at UCL, which is an older university, you can mix with more people and from a much wider range of disciplines and, anecdotally, it is more sociable than LSE, although maybe not as sociable as KCL. To compare with KCL, LSE were providing 59% of the Law trainees that KCL were despite LSE only having about 50% of the Law students of KCL, making LSE (and UCL) 15% more successful (100 - (50/59) than KCL for this.
LSE accepts a higher percentage of international students than UCL, which has a higher percentage than KCL. And this happens to be in inverse proportion to the size of their law schools, so KCL has the highest number of home students, then UCL, then LSE.