#JavaScript, 75 bytes

Rejection-based:

do x=(r=()=>4*Math.random()-2)(),y=r()
while(x*x+y*y>1)
alert(`(${[x,y]})`)

Direct method (80 bytes):

alert(`(${[(z=(m=Math).sqrt((r=m.random)()))*m.sin(p=m.PI*2*r()),z*m.cos(p)]})`)

JavaScript, 75 bytes

Rejection-based:

do x=(r=()=>4*Math.random()-2)(),y=r()
while(x*x+y*y>1)
alert(`(${[x,y]})`)

Direct method (80 bytes):

alert(`(${[(z=(m=Math).sqrt((r=m.random)()))*m.sin(p=m.PI*2*r()),z*m.cos(p)]})`)

#JavaScript, 75 bytes

Rejection-based:

do x=(r=()=>4*Math.random()-2)(),y=r()
while(x*x+y*y>1)
alert(`(${[x,y]})`)

Direct method (81 bytes):

alert(`(${[(z=(m=Math).sqrt((r=m.random)()))*m.sin(p=m.PI*2*r()),z*m.cos(p)]})`);

#JavaScript, 75 bytes

Rejection-based:

do x=(r=()=>4*Math.random()-2)(),y=r()
while(x*x+y*y>1)
alert(`(${[x,y]})`)

Direct method (80 bytes):

alert(`(${[(z=(m=Math).sqrt((r=m.random)()))*m.sin(p=m.PI*2*r()),z*m.cos(p)]})`)

#JavaScript, 75 bytes

do x=(r=()=>4*Math.random()-2)(),y=r()
while(x*x+y*y>1)
alert(`(${[x,y]})`)

Rejection-based: chooses random points in the unit square, and rejects those outside the disc.

#JavaScript, 75 bytes

Rejection-based:

do x=(r=()=>4*Math.random()-2)(),y=r()
while(x*x+y*y>1)
alert(`(${[x,y]})`)

Direct method (81 bytes):

alert(`(${[(z=(m=Math).sqrt((r=m.random)()))*m.sin(p=m.PI*2*r()),z*m.cos(p)]})`);