Naturally random devices that exploit ambient thermal noise have recently attracted attention as hardware primitives for accelerating probabilistic computing applications. One such approach is to use a low barrier nanomagnet as the free layer of a magnetic tunnel junction (MTJ), the magnetic fluctuations of which are converted to resistance fluctuations in the presence of a stable fixed layer. Here, we propose and theoretically analyze a MTJ with no fixed layers but two free layers that are circularly shaped disk magnets. We use an experimentally benchmarked model that accounts for finite-temperature magnetization dynamics, bias-dependent charge, and spin-polarized currents as well as the dipolar coupling between the free layers. We obtain analytical results for statistical averages of fluctuations that are in good agreement with the numerical model. We find that the free layers with low diameters fluctuate to randomize the resistance of the MTJ in an approximately bias-independent manner. We show how such MTJs can be used to build a binary stochastic neuron (or a p-bit) in hardware. Unlike earlier stochastic MTJs that need to operate at a specific bias point to produce random fluctuations, the proposed design can be random for a wide range of bias values, independent of spin-transfer-torque pinning. Moreover, in the absence of a carefully optimized stabled fixed layer, the symmetric double-free-layer stack can be manufactured using present-day magnetoresistive random-access memory (MRAM) technology by minimal changes to the fabrication process. Such devices can be used as hardware accelerators in energy-efficient computing schemes that require a large throughput of tunably random bits.