Whilst it's not the most common mineral associated with weight loss, in recent years there have been several studies which investigated the impact of calcium intake on body weight and fat loss.
The connection between calcium intake and weight loss was first observed in animal studies over a decade ago. Since then, several clinical trials have been conducted to investigate this relationship in humans. Interestingly there have been some positive results. Added to which there have been some correlation studies which suggest obesity rates are higher in people who don't get enough calcium.
However, one of the more often cited studies combined vitamin d3 with calcium supplementation, the problem with this study is that vitamin d deficiency is known to increase grehlin production (the hunger hormone), which for obvious reasons could skew the outcome. This could also be an issue for correlation studies, as low calcium levels can prevent effective vitamin d absorption.
That's not to say all of the studies which have shown an link to Calcium intake and weight loss are completely invalid of course, and there is still some evidence that's worth entertaining.
This shouldn't be confused with Calcium-D-Glucarate weight loss claims, which is a recent health fad. CDG works by preventing the enzyme beta-glucuronide from inhibiting the excretion of toxins, allowing them to be safely eliminated from the body. This is the root of these claims, and is as of yet completely untested in human trials and whilst it raised metabolic rate in mice, that's not something that should lead to any serious claims for humans.
In Short: There is reasonable evidence to suggest that calcium supplementation in individuals deficient in the mineral could aid weight loss, but it's not overly likely to do much by itself for most people. There are a host of secondary reasons beyond direct mechanisms that calcium deficiency could be a problem for people trying to lose weight, for example, it disrupts sleep, and even one day of disturbed sleep can spike hunger hormones dramatically. As for people without a pre existing calcium deficit, there is some evidence to suggest it could be beneficial, but not nearly as much.
How Could Caclium Help Weight Loss?
Whilst it's still not particularly well understood, one theory suggests that low calcium intake stimulates the release of certain hormones and substances that promote fat storage and inhibit fat breakdown. By increasing calcium intake, it is believed that these hormonal imbalances can be corrected, leading to enhanced weight loss.
Another proposed mechanism involves the formation of nonabsorbed complexes of calcium and fat in the intestine making it a lipase inhibitor (in short calcium can bind to fat). These complexes are then excreted in the feces, resulting in reduced overall fat absorption and weight loss.
Additionally, there are some theories that calcium may help suppress appetite, leading to reduced food intake. However, it's also worth noting, that as we state in the intro, low calcium levels can impact sleep, sleep then in turn can impact hunger and so on. Deficiencies ultimately have knock on effects.
The Most Well Known Studies on Calciums Effect on Weight Loss
Study 1: Calcium Supplementation and Body Weight
In a pooled analysis of three separate 25-week randomized controlled trials, researchers examined the effects of calcium supplementation on body weight and fat mass in premenopausal and postmenopausal women. The trials included 1000 mg/d calcium supplementation and compared it to a placebo.
The results of the analysis showed no significant differences in body weight or fat mass change between the placebo and calcium-supplemented groups. Additionally, there were no significant interactions between calcium supplementation and menopausal/diet status. These findings suggest that calcium supplementation did not significantly affect the amount of weight or fat lost by women following a moderately restricted diet for 25 weeks. 
Study 2: Calcium and Vitamin D Supplementation in Very-Low Calcium Consumers
Another study focused on individuals with very low calcium intake (less than 600 mg/day), aiming to evaluate the effects of combined calcium and vitamin D3 supplementation on body weight and fat loss. The study recruited obese and overweight subjects with a lower body mass index (BMI) cutoff. 
The results showed that calcium and vitamin D3 supplementation for 12 weeks facilitated fat loss in individuals with very low calcium intake, independent of energy restriction. The supplementation led to a significant augmentation of body fat loss and visceral fat loss compared to the control group.
Evidence Against Calcium for Weight Loss
However, not all studies have shown significant effects of calcium supplementation on weight loss. Some trials did not find a significant difference in weight or fat mass between the calcium-supplemented group and the control group. These conflicting results highlight the need for further research to better understand the relationship between calcium intake and weight loss.
Further Research Findings on Calcium and Weight
Calcium Intake and Body Weight
Several other studies have reported an inverse association between calcium intake and body weight. National Health and Nutrition Examination Surveys have provided cross-sectional evidence for this association. Retrospective analysis of data sets including calcium intake information also supports this inverse relationship.  On top of this a few studies have reported high calcium intake reduces the risks of conditions such as hypertension, diabetes, and insulin resistance often associated with obesity. 
Things That Can Impact Calciums Effectiveness
Baseline Calcium Intake
One factor that may influence the effectiveness of calcium supplementation is the baseline calcium intake of individuals. Studies have suggested that individuals with very low calcium intake may experience greater benefits from calcium supplementation compared to those with higher baseline intake levels. 
The source of calcium may also play a role in its effects on weight loss. Some studies have suggested that dairy-derived calcium may have a greater impact on weight management compared to elemental calcium supplementation. Other bioactive compounds in dairy products, such as whey-derived peptides, may contribute to these effects.
Gender may also play a role in the effects of calcium on body weight and fat loss. Some studies have shown that the association between calcium intake and obesity-related outcomes differs between men and women. For example, dairy product consumption has been associated with reduced risks of obesity and metabolic syndrome in Korean women but not in men. 
The Potential of Calcium in Weight Loss
While the results of studies exploring the relationship between calcium intake and weight loss are not entirely consistent, there is evidence to suggest that calcium supplementation may have a beneficial effect on body weight and fat loss, particularly in individuals with very low calcium intake. Although it probably doesn't belong in any mass market weight loss aids. However, more research is needed to fully understand the mechanisms behind calcium's influence on weight management.
It is important to note that calcium supplementation alone is unlikely to be a magic solution for weight loss.
6 - https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6627166/