In 1PL model, zero ability equals average ability, or change accuracy? In a test evaluation scenario, latent variable analyses are used to represent the ability of the test-taker and the difficulty of the question. A 1PL model uses a latent, random log-odds threshold, $\alpha$ to gauge the likelihood of correct response by a test taker. This is called a 1PL model.
In this 1PL dichotomous model, where $\alpha = 1$, the zero ability people are the people who only have chance accuracy, or the average ability people (who could have very high accuracy)?
 A: Assuming that your 1PL definition is 
$$P(x = 1 | \theta, \alpha) = \frac{1}{1 + \exp{[-1\cdot (\theta - \alpha)}]}$$
then no, when $\theta = 0$ and $\alpha = 1$, $P(x = 1 | \theta, \alpha) \ne 0.5$. 
The form of $P(x = 1 | \theta, \alpha) = 0.5$, commonly referred to as the inflection point, occurs only when $\theta = \alpha$; in other words, when the difficulty of the item matches the ability of the participant. The is generally why the $\alpha$ parameters are referred to as 'difficulty parameters', because larger $\alpha$ values clearly require higher ability values before the probability of positive endorsement becomes close to 1. 
A: The ability parameter zero corresponds to the 50% probability of a correct answer for the average difficult item.
This is the standard notation of a one parameter logistic model (1PL) in the item response theory (IRT) framework:
(1)  $P(x_{ij} = 1|\theta_i, \beta_j) = \frac{exp(\theta_i − \beta_j)}{1 + exp(\theta_i−\beta_j)}$
The probability of answering an item is the combination of two independent forces, the subject ability ($\theta$) and item difficulty ($\beta$).
The inclusion of a different discrimination parameter per item ($\alpha_i$) leads to a two parameter logistic model (2PL):
(2)  $P(x_{ij} = 1|\theta_i, \beta_j) = \frac{exp[\alpha_i(\theta_i − \beta_j)]}{1 + exp[\alpha_i(\theta_i − \beta_j)]}$
The 1PL is a 2PL with all the discrimination parameters are set to 1 ($\alpha_i$ = 1).
If $\theta$ and $\beta$ are zero in equation 1, their difference is zero, thus the probability of getting item right is 0.5.
This happens when the location of the Item Characteristic Curve (ICC) is zero and responder has ability equals to zero, which is the situation of item q2 in the following figure (from STATA ITEM Response THEORY REFERENCE Manual RELEASE 15).

