The answer to this question is necessarily unknowable, locked as it is inside the minds of people about to die. But the number must be substantial, and those who favor coercive suicide prevention methods often refer to the idea of insincere suicides - people who attempt suicide only to send a signal for help - to justify their preferred policies of intervention in all cases. It is only in keeping with the principal of autonomy to do what the suicide attempter wishes, they argue, and what most of them wish is to be rescued. The National Right to Life Committee puts it this way:
It is not actually a desire to die, but rather the desire to accomplish something by the attempt that drives the attempter to consider such a drastic option. Suicide is the means, not the end.
Often, suicide attempters are apparently seeking to establish some means of communication with significant persons in their lives or to test those persons' care and affection. Psychologists have concluded that other motives for attempting suicide include retaliatory abandonment (responding to a perceived abandonment by others with a revengeful "abandonment" of them through death), aggression turned inward, a search for control, manipulative guilt, punishment, escapism, frustration, or an attempt to influence someone else. Communication of these feelings -- rather than death -- is the true aim of the suicide attempter. This explains why, paradoxically but truthfully, many say after an obvious suicide attempt that they really didn't want to kill themselves. [Citations omitted.]
Certainly, there are insincere suicides - those who use a suicide attempt as a signal that they need help - in addition to sincere suicides, those whose only wish is to die. I will argue that, paradoxically, a policy of intervention and "rescue" for suicide attempters, and a general prohibition on medically assisted suicide, are actually the worst, most harmful possible policies in their effects toward insincere suicide attempters who merely wish to send a signal. These policies ensure that the signal is reliable and effective, thereby encouraging people to "communicate their feelings" through a suicide attempt rather than through more healthy methods. Coupled with widespread ignorance about the lethality of various methods, this means that many people harm themselves and even die when they do not really wish to. A general, well-publicized policy of non-intervention, or at least the possibility to opt out of intervention, coupled with a legal assisted suicide option, would actually discourage insincere suicides from attempting suicide by destroying the effectiveness of the signal of attempted suicide, and removing the perceived benefits (the rescue fantasy) that attempted suicide is currently seen to provide.
In order for a person to send a reliable signal, the suicide attempt must appear lethal while not actually being lethal. If medically assisted suicide were legally and practically available, there would be very little value in choosing any other method, and any other method would be less lethal than the medical option. (If organ donation were available as part of the medical procedure, any other method must also, incidentally, be seen as selfish.) This would interfere with the apparent lethality communicated by a suicide attempt, thereby decreasing the motivation to make a "signal" attempt in the first place.
What an insincere suicide attempter - a "signaler" - really wants is to be rescued. That is, he wants to be forcibly prevented from committing suicide, because he does not really want to commit suicide. Remove the possibility for rescue, and you remove this insincere suicide's motivation to make the potentially harmful attempt in the first place.
An analogy can be made to fights that break out on the popular television show Jerry Springer. Security personnel constantly break up fights and prevent participants from injuring each other - which causes more attacks, because participants feel they can reliably signal their "toughness," without putting themselves in danger, because of the policy of intervention (rescue) by the show's security staff. Remove the possibility for intervention, and participants would likely conduct themselves in a much less aggressive manner, as they do on other talk shows.
Of course, it might be argued that suicide intervention is justified in the case of people who are not rationally capable of making the decision to die - for instance, someone experiencing hallucinations, someone in an acute confusional state due to diabetes, or a small child (though we must be aware that people under the age of 18 and people with thought disorders and developmental disabilities still often respond to rational incentive structures, and setting up a structure that rewards them for harming themselves could itself be a cause of harm). There are two options that would at once remove the incentive for "signal" attempted suicides and protect incompetent people. One is to only allow intervention in a suicide attempt if there is reliable evidence - a judicial finding of incompetence, or underage status - that the attempter is incompetent. (Currently, the policy is to intervene in all cases, no matter what, even if reliable evidence of intent to die and competence is available.) A second option is to allow a legally effective "opt-out" procedure, so that a competent adult could legally execute a document refusing intervention in case of a suicide attempt. This option forces a choice to the would-be signaler: either execute the document, in which case one would give up the hope of rescue, hardly an option an insincere suicide would choose, or fail to execute the document, destroying the effectiveness of the signal he's trying to express.
In addition to the policies I've outlined above, accurate information about the lethality of various methods is necessary to prevent accidental death by people making insincere suicide attempts. For instance, it is difficult to say whether Megan Meier, when she hanged herself, knew that hanging has a lethality rate of 70%, and can be lethal within minutes - or that, when not lethal, it often results in permanent brain damage. Had she known this - and if she intended her behavior as a suicidal gesture, rather than intending to die, which is not known - she might have chosen a less lethal method of expressing her feelings. There is a great deal of evidence that many people do not understand the lethality of hanging asphyxiation, as evidence by the apparently accidental deaths from the "Choking Game," which kills several children every year. Paradoxically, better access to suicide information might actually save the lives of people wishing to send a signal with a suicide attempt but not to die.
As I've outlined above, a general policy of disallowing medically-assisted suicide, coupled with a policy of "rescuing" suicide attempters, is harmful and cruel not only to those who wish to die, but to those who do not. It encourages people to attempt suicide when they do not wish to die, but merely wish to send a signal, and contributes to the dangerous fantasy of rescue.
Posts by the intelligent, compassionate Dr. Maurice Bernstein at the Bioethics Discussion Blog, here and here, helped me clarify my thinking on this. Dr. Bernstein explains some of the difficulties facing emergency room doctors when faced with a patient who has attempted suicide and refuses medical intervention.